mmtheorems38 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3701-3800 - Page 38 of 123

Type	Label	Description
Statement
Theorem	fnco 3701	Composition of two functions.
|- ((F Fn A /\ G Fn B /\ ran G (_ A) -> (F o. G) Fn B)
Theorem	fnresdm 3702	A function does not change when restricted to its domain.
|- (F Fn A -> (F |` A) = F)
Theorem	fnresdisj 3703	A function restricted to a class disjoint with its domain is empty.
|- (F Fn A -> ((A i^i B) = (/) <-> (F |` B) = (/)))
Theorem	2elresin 3704	Membership in two functions restricted by each other's domain.
|- ((F Fn A /\ G Fn B) -> ((<.x, y>. e. F /\ <.x, z>. e. G) <-> (<.x, y>. e. (F |` (A i^i B)) /\ <.x, z>. e. (G |` (A i^i B)))))
Theorem	fnssresb 3705	Restriction of a function with a subclass of its domain.
|- (F Fn A -> ((F |` B) Fn B <-> B (_ A))
Theorem	fnssres 3706	Restriction of a function with a subclass of its domain.
|- ((F Fn A /\ B (_ A) -> (F |` B) Fn B)
Theorem	fnresin1 3707	Restriction of a function's domain with an intersection.
|- (F Fn A -> (F |` (A i^i B)) Fn (A i^i B))
Theorem	fnresin2 3708	Restriction of a function's domain with an intersection.
|- (F Fn A -> (F |` (B i^i A)) Fn (B i^i A))
Theorem	fnresi 3709	Functionality and domain of restricted identity.
|- (I |` A) Fn A
Theorem	fnima 3710	The image of a function's domain is its range.
|- (F Fn A -> (F"A) = ran F)
Theorem	fn0 3711	A function with empty domain is empty.
|- (F Fn (/) <-> F = (/))
Theorem	funimadisj 3712	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 3713	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 3681.
|- ((F Fn A /\ A e. B) -> F e. V)
Theorem	funex 3714	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 3713. (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 3715	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 3716	Existence of a function expressed as class of ordered pairs.
|- A e. V   =>   |- {<.x, y>. | (x e. A /\ y = B)} e. V
Theorem	opabex2g 3717	Existence of a function expressed as class of ordered pairs.
|- (A e. C -> {<.x, y>. | (x e. A /\ y = B)} e. V)
Theorem	fopabex2 3718	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	iunfopab 3719	Two ways to express a function as a class of ordered pairs.
|- B e. V   =>   |- U_x e. A {<.x, B>.} = {<.x, y>. | (x e. A /\ y = B)}
Theorem	funrnex 3720	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 3714.
|- (dom F e. B -> (Fun F -> ran F e. V))
Theorem	zfrep6 3721	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 2777 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 2767.
|- (A.x e. z E!yph -> E.wA.x e. z E.y e. w ph)
Theorem	fnopabg 3722	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 3723	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 3724	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 3725	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 3726	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 3727	Equality theorem for functions.
|- (F = G -> (F:A-->B <-> G:A-->B))
Theorem	feq2 3728	Equality theorem for functions.
|- (A = B -> (F:A-->C <-> F:B-->C))
Theorem	feq3 3729	Equality theorem for functions.
|- (A = B -> (F:C-->A <-> F:C-->B))
Theorem	feq23 3730	Equality theorem for functions. (Contributed by FL, 14-Jul-2007.)
|- ((A = C /\ B = D) -> (F:A-->B <-> F:C-->D))
Theorem	feq1d 3731	Equality deduction for mappings.
|- (ph -> F = G)   =>   |- (ph -> (F:A-->B <-> G:A-->B))
Theorem	hbf 3732	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 3733	Eliminate a mapping hypothesis for the weak deduction theorem dedth 2437, 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 3734	A mapping is a function.
|- (F:A-->B -> F Fn A)
Theorem	dffn2 3735	Any function is a mapping into V.
|- (F Fn A <-> F:A-->V)
Theorem	ffun 3736	A mapping is a function.
|- (F:A-->B -> Fun F)
Theorem	frel 3737	A mapping is a relation.
|- (F:A-->B -> Rel F)
Theorem	fdm 3738	The domain of a mapping.
|- (F:A-->B -> dom F = A)
Theorem	fdmi 3739	The domain of a mapping.
|- F:A-->B   =>   |- dom F = A
Theorem	frn 3740	The range of a mapping.
|- (F:A-->B -> ran F (_ B)
Theorem	dffn3 3741	A function maps to its range.
|- (F Fn A <-> F:A-->ran F)
Theorem	fss 3742	Expanding the codomain of a mapping.
|- ((F:A-->B /\ B (_ C) -> F:A-->C)
Theorem	fco 3743	Composition of two mappings.
|- ((F:B-->C /\ G:A-->B) -> (F o. G):A-->C)
Theorem	fssxp 3744	A mapping is a class of ordered pairs.
|- (F:A-->B -> F (_ (A X. B))
Theorem	funssxp 3745	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 3746	A mapping is a partial function.
|- (F:A-->B -> (F:dom F-->B /\ dom F (_ A))
Theorem	opelf 3747	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 3748	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 3749	Composition of two functions.
|- ((F Fn A /\ G:B-->A) -> (F o. G) Fn B)
Theorem	fssres 3750	Restriction of a function with a subclass of its domain.
|- ((F:A-->B /\ C (_ A) -> (F |` C):C-->B)
Theorem	fssres2 3751	Restriction of a restricted function with a subclass of its domain.
|- (((F |` A):A-->B /\ C (_ A) -> (F |` C):C-->B)
Theorem	fcoi1 3752	Composition of a mapping and restricted identity.
|- (F:A-->B -> (F o. (I |` A)) = F)
Theorem	fcoi2 3753	Composition of restricted identity and a mapping.
|- (F:A-->B -> ((I |` B) o. F) = F)
Theorem	feu 3754	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 3755	The converse of a restriction of a function.
|- (F:A-->B -> `'(F |` A) = (`'F |` B))
Theorem	fimacnvdisj 3756	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 3757	Function into an intersection.
|- B =/= (/)   =>   |- (F:A-->|^|B <-> A.x e. B F:A-->x)
Theorem	fin 3758	Mapping into an intersection.
|- (F:A-->(B i^i C) <-> (F:A-->B /\ F:A-->C))
Theorem	fex 3759	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 3760	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 3761	Existence of a set of functions.
|- A e. V   &   |- B e. V   &   |- F = {x | (x:A-->B /\ ph)}   =>   |- F e. V
Theorem	dmfex 3762	If a mapping is a set, its domain is a set.
|- ((F e. C /\ F:A-->B) -> A e. V)
Theorem	f0 3763	The empty function.
|- (/):(/)-->A
Theorem	f00 3764	A class is a function with empty codomain iff it and its domain are empty.
|- (F:A-->(/) <-> (F = (/) /\ A = (/)))
Theorem	fconst 3765	A cross product with a singleton is a constant function.
|- B e. V   =>   |- (A X. {B}):A-->{B}
Theorem	fconstg 3766	A cross product with a singleton is a constant function.
|- (B e. C -> (A X. {B}):A-->{B})
Theorem	f1eq1 3767	Equality theorem for one-to-one functions.
|- (F = G -> (F:A-1-1->B <-> G:A-1-1->B))
Theorem	f1eq2 3768	Equality theorem for one-to-one functions.
|- (A = B -> (F:A-1-1->C <-> F:B-1-1->C))
Theorem	f1eq3 3769	Equality theorem for one-to-one functions.
|- (A = B -> (F:C-1-1->A <-> F:C-1-1->B))
Theorem	hbf1 3770	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	dff12 3771	Alternate definition of a one-to-one function.
|- (F:A-1-1->B <-> (F:A-->B /\ A.yE*x xFy))
Theorem	f1f 3772	A one-to-one mapping is a mapping.
|- (F:A-1-1->B -> F:A-->B)
Theorem	f1cnv 3773	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 3774	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)
Theorem	foeq1 3775	Equality theorem for onto functions.
|- (F = G -> (F:A-onto->B <-> G:A-onto->B))
Theorem	foeq2 3776	Equality theorem for onto functions.
|- (A = B -> (F:A-onto->C <-> F:B-onto->C))
Theorem	foeq3 3777	Equality theorem for onto functions.
|- (A = B -> (F:C-onto->A <-> F:C-onto->B))
Theorem	hbfo 3778	Bound-variable hypothesis builder for an onto 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-onto->B -> A.x F:A-onto->B)
Theorem	fof 3779	An onto mapping is a mapping.
|- (F:A-onto->B -> F:A-->B)
Theorem	fofun 3780	An onto mapping is a function.
|- (F:A-onto->B -> Fun F)
Theorem	fofn 3781	An onto mapping is a function on its domain.
|- (F:A-onto->B -> F Fn A)
Theorem	forn 3782	The codomain of an onto function is its range.
|- (F:A-onto->B -> ran F = B)
Theorem	dffo2 3783	Alternate definition of an onto function.
|- (F:A-onto->B <-> (F:A-->B /\ ran F = B))
Theorem	foima 3784	The image of the domain of an onto function.
|- (F:A-onto->B -> (F"A) = B)
Theorem	dffn4 3785	A function maps onto its range.
|- (F Fn A <-> F:A-onto->ran F)
Theorem	funforn 3786	A function maps its domain onto its range.
|- (Fun A <-> A:dom A-onto->ran A)
Theorem	fornex 3787	If the domain of an onto function exists, so does its codomain.
|- (A e. C -> (F:A-onto->B -> B e. V))
Theorem	fodmrnu 3788	An onto function has unique domain and range.
|- ((F:A-onto->B /\ F:C-onto->D) -> (A = C /\ B = D))
Theorem	fores 3789	Restriction of a function.
|- ((Fun F /\ A (_ dom F) -> (F |` A):A-onto->(F"A))
Theorem	foco 3790	Composition of onto functions.
|- ((F:B-onto->C /\ G:A-onto->B) -> (F o. G):A-onto->C)
Theorem	foconst 3791	A non-zero constant function is onto.
|- ((F:A-->{B} /\ F =/= (/)) -> F:A-onto->{B})
Theorem	f1oeq1 3792	Equality theorem for one-to-one onto functions.
|- (F = G -> (F:A-1-1-onto->B <-> G:A-1-1-onto->B))
Theorem	f1oeq2 3793	Equality theorem for one-to-one onto functions.
|- (A = B -> (F:A-1-1-onto->C <-> F:B-1-1-onto->C))
Theorem	f1oeq3 3794	Equality theorem for one-to-one onto functions.
|- (A = B -> (F:C-1-1-onto->A <-> F:C-1-1-onto->B))
Theorem	hbf1o 3795	Bound-variable hypothesis builder for a one-to-one onto 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-onto->B -> A.x F:A-1-1-onto->B)
Theorem	f1of1 3796	A one-to-one onto mapping is a one-to-one mapping.
|- (F:A-1-1-onto->B -> F:A-1-1->B)
Theorem	f1of 3797	A one-to-one onto mapping is a mapping.
|- (F:A-1-1-onto->B -> F:A-->B)
Theorem	f1ofn 3798	A one-to-one onto mapping is function on its domain.
|- (F:A-1-1-onto->B -> F Fn A)
Theorem	f1ofun 3799	A one-to-one onto mapping is a function.
|- (F:A-1-1-onto->B -> Fun F)
Theorem	f1orel 3800	A one-to-one onto mapping is a relation.
|- (F:A-1-1-onto->B -> Rel F)