mmtheorems33 - Metamath Proof Explorer

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10774

Statement List for Metamath Proof Explorer - 3201-3300 - Page 33 of 108

Type	Label	Description
Statement
Definition	df-fun 3201	Define a function. Definition 10.1 of [Quine] p. 65. For alternate definitions, see dffun2 3535, dffun3 3536, dffun4 3537, dffun5 3538, dffunmo 3540, dffun6 3548, and dffun7 3549.
|- (Fun A <-> (Rel A /\ (A o. `'A) (_ I))
Definition	df-fn 3202	Define a function with domain. Definition 6.15(1) of [TakeutiZaring] p. 27.
|- (A Fn B <-> (Fun A /\ dom A = B))
Definition	df-f 3203	Define a function (mapping) with domain and codomain. Definition 6.15(3) of [TakeutiZaring] p. 27.
|- (F:A-->B <-> (F Fn A /\ ran F (_ B))
Definition	df-f1 3204	Define a one-to-one function. For an equivalent definition see f11 3673. Compare Definition 6.15(5) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow).
|- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
Definition	df-fo 3205	Define an onto function. Definition 6.15(4) of [TakeutiZaring] p. 27. We use their notation ("onto" under the arrow).
|- (F:A-onto->B <-> (F Fn A /\ ran F = B))
Definition	df-f1o 3206	Define a one-to-one onto function. For equivalent definitions see f1o2 3702, f1o3 3703, f1o4 3705, and f1o5 3706. Compare Definition 6.15(6) of [TakeutiZaring] p. 27. We use their notation ("1-1" above the arrow and "onto" below the arrow).
|- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ F:A-onto->B))
Definition	df-fv 3207	Define the value of a function. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 3437), our definition apparently does not appear in the literature; but it is quite convenient: it can be applied to any class and evaluates to the empty set when it is not meaningful (as shown by ndmfv 3754 and fvprc 3730). The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. It means the same thing as the more familiar F(A) notation for a function's value at A, i.e. "F of A," but without context-dependent ambiguity. For conventional alternate definitions (that fail to evaluate to the empty set for proper classes), see fv2 3729 and fv3 3742. Restricted equivalents that require F to be a function are shown in funfv 3779 and funfv2 3780. For the familiar definition of function value in terms of ordered pair membership, see funopfvb 3765.
|- (F` A) = U.{x | (F"{A}) = {x}}
Definition	df-iso 3208	Define the isomorphism predicate. We read this as "H is an R, S isomorphism of A onto B." Normally, R and S are ordering relations on A and B respectively. Definition 6.28 of [TakeutiZaring] p. 32, whose notation is the same as ours except that R and S are subscripts.
|- (H Isom R, S (A, B) <-> (H:A-1-1-onto->B /\ A.x e. A A.y e. A (xRy <-> (H` x)S(H` y))))
Theorem	xpeq1 3209	Equality theorem for cross product.
|- (A = B -> (A X. C) = (B X. C))
Theorem	xpeq2 3210	Equality theorem for cross product.
|- (A = B -> (C X. A) = (C X. B))
Theorem	elxp 3211	Membership in a cross product.
|- (A e. (B X. C) <-> E.xE.y(A = <.x, y>. /\ (x e. B /\ y e. C)))
Theorem	elxp2 3212	Membership in a cross product.
|- (A e. (B X. C) <-> E.x e. B E.y e. C A = <.x, y>.)
Theorem	hbxp 3213	Bound-variable hypothesis builder for cross product.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. (A X. B) -> A.x y e. (A X. B))
Theorem	opelxp1 3214	The first member of an ordered pair of classes in a cross product belongs to first cross product argument.
|- (<.A, B>. e. (C X. D) -> A e. C)
Theorem	otelxp1 3215	The first member of an ordered triple of classes in a cross product belongs to first cross product argument.
|- (<.<.A, B>., C>. e. ((R X. S) X. T) -> A e. R)
Theorem	brrelex 3216	A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.)
|- ((Rel R /\ ARB) -> A e. V)
Theorem	brrelexi 3217	The first argument of a binary relation exists. (An artifact of our ordered pair definition.)
|- Rel R   =>   |- (ARB -> A e. V)
Theorem	nprrel 3218	No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)
|- Rel R   &   |- -. A e. V   =>   |- -. ARB
Theorem	fconstopab 3219	Representation of a constant function using ordered pairs.
|- (A X. {B}) = {<.x, y>. | (x e. A /\ y = B)}
Theorem	vtoclr 3220	Variable to class conversion of transitive relation.
|- Rel R   &   |- ((xRy /\ yRz) -> xRz)   =>   |- (C e. D -> ((ARB /\ BRC) -> ARC))
Theorem	vtoclrbr 3221	Variable to class conversion of transitive, reflexive relation.
|- Rel R   &   |- ((xRy /\ yRz) -> xRz)   &   |- xRx   =>   |- ((ARB /\ BRC) -> ARC)
Theorem	vtoclibr 3222	Variable to class conversion of transitive, irreflexive relation.
|- Rel R   &   |- ((xRy /\ yRz) -> xRz)   &   |- -. xRx   =>   |- ((ARB /\ BRC) -> ARC)
Theorem	opelxp 3223	Ordered pair membership in a cross product.
|- B e. V   =>   |- (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D))
Theorem	brxp 3224	Binary relation on a cross product.
|- B e. V   =>   |- (A(C X. D)B <-> (A e. C /\ B e. D))
Theorem	opelxpg 3225	Ordered pair membership in a cross product.
|- (B e. R -> (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D)))
Theorem	opelxpi 3226	Ordered pair membership in a cross product (implication).
|- ((A e. C /\ B e. D) -> <.A, B>. e. (C X. D))
Theorem	ralxp 3227	Universal quantification restricted to a cross product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution.
|- (x = <.y, z>. -> (ph <-> ps))   =>   |- (A.x e. (A X. B)ph <-> A.y e. A A.z e. B ps)
Theorem	rexxp 3228	Existential quantification restricted to a cross product is equivalent to a double restricted quantification.
|- (x = <.y, z>. -> (ph <-> ps))   =>   |- (E.x e. (A X. B)ph <-> E.y e. A E.z e. B ps)
Theorem	ralxpf 3229	Version of ralxp 3227 with bound-variable hypotheses.
|- (ph -> A.yph)   &   |- (ph -> A.zph)   &   |- (ps -> A.xps)   &   |- (x = <.y, z>. -> (ph <-> ps))   =>   |- (A.x e. (A X. B)ph <-> A.y e. A A.z e. B ps)
Theorem	opthprc 3230	Justification theorem for an ordered pair definition that works for any classes, including proper classes. This is a possible definition implied by the footnote in [Jech] p. 78, which says, "The sophisticated reader will not object to our use of a pair of classes."
|- (((A X. {(/)}) u. (B X. {{(/)}})) = ((C X. {(/)}) u. (D X. {{(/)}})) <-> (A = C /\ B = D))
Theorem	brelg 3231	Two things in a binary relation belong to the relation's domain.
|- R (_ (C X. D)   =>   |- (B e. S -> (ARB -> (A e. C /\ B e. D)))
Theorem	brel 3232	Membership in superset of binary relation.
|- B e. V   &   |- R (_ (C X. D)   =>   |- (ARB -> (A e. C /\ B e. D))
Theorem	elxp3 3233	Membership in a cross product.
|- (A e. (B X. C) <-> E.xE.y(<.x, y>. = A /\ <.x, y>. e. (B X. C)))
Theorem	xpundi 3234	Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52.
|- (A X. (B u. C)) = ((A X. B) u. (A X. C))
Theorem	xpundir 3235	Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52.
|- ((A u. B) X. C) = ((A X. C) u. (B X. C))
Theorem	xpun 3236	The cross product of two unions.
|- ((A u. B) X. (C u. D)) = (((A X. C) u. (A X. D)) u. ((B X. C) u. (B X. D)))
Theorem	elvv 3237	Membership in universal class of ordered pairs.
|- (A e. (V X. V) <-> E.xE.y A = <.x, y>.)
Theorem	elvvuni 3238	An ordered pair contains its union.
|- (A e. (V X. V) -> U.A e. A)
Theorem	xpss 3239	A cross product is included in the ordered pair universe. Exercise 3 of [TakeutiZaring] p. 25.
|- (A X. B) (_ (V X. V)
Theorem	brinxp2 3240	Intersection of binary relation with cross product.
|- (B e. S -> (A(R i^i (C X. D))B <-> (A e. C /\ B e. D /\ ARB)))
Theorem	brinxp 3241	Intersection of binary relation with cross product.
|- ((A e. C /\ B e. D) -> (ARB <-> A(R i^i (C X. D))B))
Theorem	weinxp 3242	Intersection of well-ordering with cross product of its field.
|- (R We A <-> (R i^i (A X. A)) We A)
Theorem	opabssxp 3243	An abstraction relation is a subset of a related cross product.
|- {<.x, y>. | ((x e. A /\ y e. B) /\ ph)} (_ (A X. B)
Theorem	optocl 3244	Implicit substitution of class for ordered pair.
|- D = (B X. C)   &   |- (<.x, y>. = A -> (ph <-> ps))   &   |- ((x e. B /\ y e. C) -> ph)   =>   |- (A e. D -> ps)
Theorem	2optocl 3245	Implicit substitution of classes for ordered pairs.
|- R = (C X. D)   &   |- (<.x, y>. = A -> (ph <-> ps))   &   |- (<.z, w>. = B -> (ps <-> ch))   &   |- (((x e. C /\ y e. D) /\ (z e. C /\ w e. D)) -> ph)   =>   |- ((A e. R /\ B e. R) -> ch)
Theorem	3optocl 3246	Implicit substitution of classes for ordered pairs.
|- R = (D X. F)   &   |- (<.x, y>. = A -> (ph <-> ps))   &   |- (<.z, w>. = B -> (ps <-> ch))   &   |- (<.v, u>. = C -> (ch <-> th))   &   |- (((x e. D /\ y e. F) /\ (z e. D /\ w e. F) /\ (v e. D /\ u e. F)) -> ph)   =>   |- ((A e. R /\ B e. R /\ C e. R) -> th)
Theorem	opbrop 3247	Ordered pair membership in a relation. Special case.
|- (((z = A /\ w = B) /\ (v = C /\ u = D)) -> (ph <-> ps))   &   |- R = {<.x, y>. | ((x e. (S X. S) /\ y e. (S X. S)) /\ E.zE.wE.vE.u(