mmtheorems37 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3601-3700 - Page 37 of 107

Type	Label	Description
Statement
Theorem	fnresi 3601	Functionality and domain of restricted identity.
|- (I |` A) Fn A
Theorem	fnima 3602	The image of a function's domain is its range.
|- (F Fn A -> (F"A) = ran F)
Theorem	fn0 3603	A function with empty domain is empty.
|- (F Fn (/) <-> F = (/))
Theorem	funimadisj 3604	A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)
|- ((F Fn A /\ (A i^i C) = (/)) -> (F"C) = (/))
Theorem	fnex 3605	If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 3573.
|- ((F Fn A /\ A e. B) -> F e. V)
Theorem	funex 3606	If the domain of a function exists, so the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 3605. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.)
|- ((Fun F /\ dom F e. B) -> F e. V)
Theorem	opabex 3607	Existence of a function expressed as class of ordered pairs.
|- A e. V   &   |- (x e. A -> E*yph)   =>   |- {<.x, y>. | (x e. A /\ ph)} e. V
Theorem	opabex2 3608	Existence of a function expressed as class of ordered pairs.
|- A e. V   =>   |- {<.x, y>. | (x e. A /\ y = B)} e. V
Theorem	opabex2g 3609	Existence of a function expressed as class of ordered pairs.
|- (A e. C -> {<.x, y>. | (x e. A /\ y = B)} e. V)
Theorem	fopabex2 3610	Existence of a function expressed as class of ordered pairs.
|- A e. V   &   |- F = {<.x, y>. | (x e. A /\ y = B)}   =>   |- F e. V
Theorem	funrnex 3611	If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 3606.
|- (dom F e. B -> (Fun F -> ran F e. V))
Theorem	zfrep6 3612	A version of the Axiom of Replacement. Normally ph would have free variables x and y. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 2701 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version in place of our ax-rep 2691.
|- (A.x e. z E!yph -> E.wA.x e. z E.y e. w ph)
Theorem	fnopabg 3613	Functionality and domain of an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ ph)}   =>   |- (A.x e. A E!yph <-> F Fn A)
Theorem	fnopab2g 3614	Functionality and domain of an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = B)}   =>   |- (A.x e. A B e. V <-> F Fn A)
Theorem	fnopab 3615	Functionality and domain of an ordered-pair class abstraction.
|- (x e. A -> E!yph)   &   |- F = {<.x, y>. | (x e. A /\ ph)}   =>   |- F Fn A
Theorem	fnopab2 3616	Functionality and domain of an ordered-pair class abstraction.
|- B e. V   &   |- F = {<.x, y>. | (x e. A /\ y = B)}   =>   |- F Fn A
Theorem	dmopab2 3617	Domain of an ordered-pair class abstraction that specifies a function.
|- B e. V   &   |- F = {<.x, y>. | (x e. A /\ y = B)}   =>   |- dom F = A
Theorem	feq1 3618	Equality theorem for functions.
|- (F = G -> (F:A-->B <-> G:A-->B))
Theorem	feq2 3619	Equality theorem for functions.
|- (A = B -> (F:A-->C <-> F:B-->C))
Theorem	feq3 3620	Equality theorem for functions.
|- (A = B -> (F:C-->A <-> F:C-->B))
Theorem	feq23 3621	Equality theorem for functions. (Contributed by FL, 14-Jul-2007.)
|- ((A = C /\ B = D) -> (F:A-->B <-> F:C-->D))
Theorem	feq1d 3622	Equality deduction for mappings.
|- (ph -> F = G)   =>   |- (ph -> (F:A-->B <-> G:A-->B))
Theorem	hbf 3623	Bound-variable hypothesis builder for a mapping.
|- (y e. F -> A.x y e. F)   &   |- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (F:A-->B -> A.x F:A-->B)
Theorem	elimf 3624	Eliminate a mapping hypothesis for the weak deduction theorem dedth 2381, when a special case G:A-->B is provable, in order to convert F:A-->B from a hypothesis to an antecedent.
|- G:A-->B   =>   |- if(F:A-->B, F, G):A-->B
Theorem	ffn 3625	A mapping is a function.
|- (F:A-->B -> F Fn A)
Theorem	fnf 3626	Any function is a mapping into V.
|- (F Fn A <-> F:A-->V)
Theorem	ffun 3627	A mapping is a function.
|- (F:A-->B -> Fun F)
Theorem	frel 3628	A mapping is a relation.
|- (F:A-->B -> Rel F)
Theorem	fdm 3629	The domain of a mapping.
|- (F:A-->B -> dom F = A)
Theorem	fdmi 3630	The domain of a mapping.
|- F:A-->B   =>   |- dom F = A
Theorem	frn 3631	The range of a mapping.
|- (F:A-->B -> ran F (_ B)
Theorem	fnfrn 3632	A function maps to its range.
|- (F Fn A <-> F:A-->ran F)
Theorem	fss 3633	Expanding the codomain of a mapping.
|- ((F:A-->B /\ B (_ C) -> F:A-->C)
Theorem	fco 3634	Composition of two mappings.
|- ((F:B-->C /\ G:A-->B) -> (F o. G):A-->C)
Theorem	fssxp 3635	A mapping is a class of ordered pairs.
|- (F:A-->B -> F (_ (A X. B))
Theorem	funssxp 3636	Two ways of specifying a partial function from A to B.
|- ((Fun F /\ F (_ (A X. B)) <-> (F:dom F-->B /\ dom F (_ A))
Theorem	ffdm 3637	A mapping is a partial function.
|- (F:A-->B -> (F:dom F-->B /\ dom F (_ A))
Theorem	opelf 3638	The members of an ordered pair element of a mapping belong to the mapping's domain and codomain.
|- D e. V   =>   |- ((F:A-->B /\ <.C, D>. e. F) -> (C e. A /\ D e. B))
Theorem	fun 3639	The union of two functions with disjoint domains.
|- (((F:A-->C /\ G:B-->D) /\ (A i^i B) = (/)) -> (F u. G):(A u. B)-->(C u. D))
Theorem	fnfco 3640	Composition of two functions.
|- ((F Fn A /\ G:B-->A) -> (F o. G) Fn B)
Theorem	fssres 3641	Restriction of a function with a subclass of its domain.
|- ((F:A-->B /\ C (_ A) -> (F |` C):C-->B)
Theorem	fssres2 3642	Restriction of a restricted function with a subclass of its domain.
|- (((F |` A):A-->B /\ C (_ A) -> (F |` C):C-->B)
Theorem	fcoi1 3643	Composition of a mapping and restricted identity.
|- (F:A-->B -> (F o. (I |` A)) = F)
Theorem	fcoi2 3644	Composition of restricted identity and a mapping.
|- (F:A-->B -> ((I |` B) o. F) = F)
Theorem	feu 3645	There is exactly one value of a function in its codomain.
|- ((F:A-->B /\ C e. A) -> E!y e. B <.C, y>. e. F)
Theorem	fcnvres 3646	The converse of a restriction of a function.
|- (F:A-->B -> `'(F |` A) = (`'F |` B))
Theorem	fimacnvdisj 3647	The pre-image of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
|- ((F:A-->B /\ (B i^i C) = (/)) -> (`'F"C) = (/))
Theorem	fint 3648	Function into an intersection.
|- B =/= (/)   =>   |- (F:A-->|^|B <-> A.x e. B F:A-->x)
Theorem	fin 3649	Mapping into an intersection.
|- (F:A-->(B i^i C) <-> (F:A-->B /\ F:A-->C))
Theorem	fex 3650	If the domain of a mapping is a set, the function is a set.
|- ((F:A-->B /\ A e. C) -> F e. V)
Theorem	fabexg 3651	Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.)
|- F = {x | (x:A-->B /\ ph)}   =>   |- ((A e. C /\ B e. D) -> F e. V)
Theorem	fabex 3652	Existence of a set of functions.
|- A e. V   &   |- B e. V   &   |- F = {x | (x:A-->B /\ ph)}   =>   |- F e. V
Theorem	dmfex 3653	If a mapping is a set, its domain is a set.
|- ((F e. C /\ F:A-->B) -> A e. V)
Theorem	f0 3654	The empty function.
|- (/):(/)-->A
Theorem	f00 3655	A class is a function with empty codomain iff it and its domain are empty.
|- (F:A-->(/) <-> (F = (/) /\ A = (/)))
Theorem	fconst 3656	A cross product with a singleton is a constant function.
|- B e. V   =>   |- (A X. {B}):A-->{B}
Theorem	fconstg 3657	A cross product with a singleton is a constant function.
|- (B e. C -> (A X. {B}):A-->{B})
Theorem	f1eq1 3658	Equality theorem for one-to-one functions.
|- (F = G -> (F:A-1-1->B <-> G:A-1-1->B))
Theorem	f1eq2 3659	Equality theorem for one-to-one functions.
|- (A = B -> (F:A-1-1->C <-> F:B-1-1->C))
Theorem	f1eq3 3660	Equality theorem for one-to-one functions.
|- (A = B -> (F:C-1-1->A <-> F:C-1-1->B))
Theorem	hbf1 3661	Bound-variable hypothesis builder for a one-to-one function.
|- (y e. F -> A.x y e. F)   &   |- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (F:A-1-1->B -> A.x F:A-1-1->B)
Theorem	f11 3662	Alternate definition of a one-to-one function.
|- (F:A-1-1->B <-> (F:A-->B /\ A.yE*x xFy))
Theorem	f1f 3663	A one-to-one mapping is a mapping.
|- (F:A-1-1->B -> F:A-->B)
Theorem	f1cnv 3664	Two ways to express that a set A (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one. We do not introduce a separate notation since we rarely use it.
|- (`'`'A:dom A-1-1->V <-> (Fun `'A /\ Fun `'`'A))
Theorem	f1co 3665	Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25.
|- ((F:B-1-1->C /\ G:A-1-1->B) -> (F o. G):A-1-1->C