mmtheorems39 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3801-3900 - Page 39 of 108

Type	Label	Description
Statement
Theorem	elrnopab 3801	Membership in the range of an ordered pair class abstraction.
|- B e. V   &   |- F = {<.x, y>. | (x e. A /\ y = B)}   =>   |- (C e. ran F <-> E.x e. A C = B)
Theorem	chfnrn 3802	The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain.
|- ((F Fn A /\ A.x e. A (F` x) e. x) -> ran F (_ U.A)
Theorem	funfvop 3803	Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41.
|- ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. F)
Theorem	fvimacnvi 3804	A member of a preimage is a function value argument.
|- ((Fun F /\ A e. (`'F"B)) -> (F` A) e. B)
Theorem	fvimacnv 3805	The argument of a function value belongs to the pre-image of any class containing the function value. (Contributed by Raph Levien, 20-Nov-2006. He remarks: "This proof is unsatisfying, because it seems to me that funimass2 3573 could probably be strengthened to a biconditional.")
|- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> A e. (`'F"B)))
Theorem	funimass3 3806	A kind of contraposition law that infers an image subclass from a subclass of a pre-image. (Contributed by Raph Levien, 20-Nov-2006. He remarks: "Likely this could be proved directly, and fvimacnv 3805 would be the special case of A being a singleton, but it works this way round too.")
|- ((Fun F /\ A (_ dom F) -> ((F"A) (_ B <-> A (_ (`'F"B)))
Theorem	funimass5 3807	A subclass of a preimage in terms of function values.
|- ((Fun F /\ A (_ dom F) -> (A (_ (`'F"B) <-> A.x e. A (F` x) e. B))
Theorem	funconstss 3808	Two ways of specifying that a function is constant on a subdomain.
|- ((Fun F /\ A (_ dom F) -> (A.x e. A (F` x) = B <-> A (_ (`'F"{B})))
Theorem	fvimacnvALT 3809	Another proof of fvimacnv 3805, based on funimass3 3806. If funimass3 3806 is ever proved directly, as opposed to using funimacnv 3571 pointwise, then the proof of funimacnv 3571 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.)
|- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> A e. (`'F"B)))
Theorem	fimacnv 3810	The pre-image of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)
|- (F:A-->B -> (`'F"B) = A)
Theorem	fnopfv 3811	Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41.
|- ((F Fn A /\ B e. A) -> <.B, (F` B)>. e. F)
Theorem	fvelrn 3812	A function's value belongs to its range.
|- ((Fun F /\ A e. dom F) -> (F` A) e. ran F)
Theorem	fnfvelrn 3813	A function's value belongs to its range.
|- ((F Fn A /\ B e. A) -> (F` B) e. ran F)
Theorem	ffvelrn 3814	A function's value belongs to its codomain.
|- ((F:A-->B /\ C e. A) -> (F` C) e. B)
Theorem	ffvelrni 3815	A function's value belongs to its codomain.
|- F:A-->B   =>   |- (C e. A -> (F` C) e. B)
Theorem	dff4 3816	Alternate definition of a mapping.
|- (F:A-->B <-> (F Fn A /\ F (_ (A X. B)))
Theorem	dff2 3817	Alternate definition of a mapping.
|- (F:A-->B <-> (F (_ (A X. B) /\ A.x e. A E!y xFy))
Theorem	dff3 3818	Alternate definition of a mapping.
|- (F:A-->B <-> (F (_ (A X. B) /\ A.x e. A E!y e. B xFy))
Theorem	dffo3 3819	An onto mapping expressed in terms of function values.
|- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x e. A y = (F` x)))
Theorem	dffo4 3820	Alternate definition of an onto mapping.
|- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x e. A xFy))
Theorem	dffo5 3821	Alternate definition of an onto mapping.
|- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x xFy))
Theorem	exfo 3822	A relation equivalent to the existence of an onto mapping. The right-hand f is not necessarily a function.
|- (E.f f:A-onto->B <-> E.f(A.x e. A E!y e. B xfy /\ A.x e. B E.y e. A yfx))
Theorem	fopab2 3823	Functionality of an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = C)}   =>   |- (A.x e. A C e. B <-> F:A-->B)
Theorem	fopabssxp 3824	Inclusion of a function in a cross product.
|- F = {<.x, y>. | (x e. A /\ y = C)}   =>   |- (A.x e. A C e. B -> F (_ (A X. B))
Theorem	rnssopab 3825	Range of a function that is expressed as an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = C)}   &   |- C e. V   =>   |- (A.x e. A C e. B <-> ran F (_ B)
Theorem	fopab3 3826	Functionality of an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = C)}   &   |- C e. V   =>   |- (ran F (_ B <-> F:A-->B)
Theorem	fopab 3827	Functionality of an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = C)}   &   |- (x e. A -> C e. B)   =>   |- F:A-->B
Theorem	ffnfv 3828	A function maps to a class to which all values belong.
|- (F:A-->B <-> (F Fn A /\ A.x e. A (F` x) e. B))
Theorem	ffnfvf 3829	A function maps to a class to which all values belong. This version of ffnfv 3828 uses bound-variable hypotheses instead of distinct variable conditions.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   &   |- (y e. F -> A.x y e. F)   =>   |- (F:A-->B <-> (F Fn A /\ A.x e. A (F` x) e. B))
Theorem	fnfvrnss 3830	An upper bound for range determined by function values.
|- ((F Fn A /\ A.x e. A (F` x) e. B) -> ran F (_ B)
Theorem	fopabfv 3831	Representation of a mapping in terms of its values.
|- (F:A-->B <-> (F = {<.x, y>. | (x e. A /\ y = (F` x))} /\ A.x e. A (F` x) e. B))
Theorem	fopabco 3832	Composition of two functions expressed as ordered-pair class abstractions. Note that v may be assigned to w, y, or z if desired.
|- R e. V   &   |- S e. V   &   |- T e. V   &   |- (z = R -> S = T)   &   |- F = {<.x, y>. | (x e. A /\ y = R)}   &   |- G = {<.z, w>. | (z e. B /\ w = S)}   &   |- H = {<.x, v>. | (x e. A /\ v = T)}   =>   |- (ran F (_ B -> (G o. F) = H)
Theorem	fopabcos 3833	Composition of two functions expressed as ordered-pair class abstractions.
|- C e. V   &   |- D e. V   &   |- F = {<.x, y>. | (x e. A /\ y = C)}   &   |- G = {<.x, y>. | (x e. B /\ y = D)}   =>   |- (ran G (_ A -> (F o. G) = {<.x, y>. | (x e. B /\ y = [_D / x]_C)})
Theorem	fsn 3834	A function maps a singleton to a singleton iff it is the singleton of a ordered pair.
|- A e. V   &   |- B e. V   =>   |- (F:{A}-->{B} <-> F = {<.A, B>.})
Theorem	xpsn 3835	The cross product of two singletons.
|- A e. V   &   |- B e. V   =>   |- ({A} X. {B}) = {<.A, B>.}
Theorem	fsn2 3836	A function that maps a singleton to a class is the singleton of an ordered pair.
|- A e. V   =>   |- (F:{A}-->B <-> ((F` A) e. B /\ F = {<.A, (F` A)>.}))
Theorem	fnressn 3837	A function restricted to a singleton.
|- ((F Fn A /\ B e. A) -> (F |` {B}) = {<.B, (F` B)>.})
Theorem	fressnfv 3838	The value of a function restricted to a singleton.
|- ((F Fn A /\ B e. A) -> ((F |` {B}):{B}-->C <-> (F` B) e. C))
Theorem	fvconst 3839	The value of a constant function.
|- ((F:A-->{B} /\ C e. A) -> (F` C) = B)
Theorem	fopabsn 3840	The singleton of an ordered pair expressed as an ordered pair class abstraction.
|- A e. V   &   |- B e. V   =>   |- {<.A, B>.} = {<.x, y>. | (x e. {A} /\ y = B)}
Theorem	fopabap 3841	Append an additional value to a function.
|- A e. V   &   |- B e. V   &   |- (R u. {A}) = S   &   |- (x = A -> C = B)   =>   |- ({<.x, y>. | (x e. R /\ y = C)} u. {<.A, B>.}) = {<.x, y>. | (x e. S /\ y = C)}
Theorem	fvi 3842	The value of the identity function.
|- (A e. B -> (I` A) = A)
Theorem	fvresi 3843	The value of a restricted identity function.
|- (B e. A -> ((I |` A)` B) = B)
Theorem	fvconst2g 3844	The value of a constant function.
|- ((B e. D /\ C e. A) -> ((A X. {B})` C) = B)
Theorem	fconst2g 3845	A constant function expressed as a cross product.
|- (B e. C -> (F:A-->{B} <-> F = (A X. {B})))
Theorem	fvconst2 3846	The value of a constant function.
|- B e. V   =>   |- (C e. A -> ((A X. {B})` C) = B)
Theorem	fconst2 3847	A constant function expressed as a cross product.
|- B e. V   =>   |- (F:A-->{B} <-> F = (A X. {B}))
Theorem	fconst5 3848	Two ways to express that a function is consta