mmtheorems48 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4701-4800 - Page 48 of 175

Type	Label	Description
Statement
Theorem	csbfvg 4701	Substitution for a function value.
|- (A e. C -> [_A / x]_(F` x) = (F` A))
Theorem	ndmfv 4702	The value of a class outside its domain is the empty set.
|- (-. A e. dom F -> (F` A) = (/))
Theorem	ndmfvrcl 4703	Reverse closure law for function with the empty set not in its domain.
|- dom F = S   &   |- -. (/) e. S   =>   |- ((F` A) e. S -> A e. S)
Theorem	elfvdm 4704	If a function value has a member, the argument belongs to the domain.
|- (A e. (F` B) -> B e. dom F)
Theorem	nfvres 4705	The value of a non-member of a restriction is the empty set.
|- (-. A e. B -> ((F |` B)` A) = (/))
Theorem	nfunsn 4706	If the restriction of a class to a singleton is not a function, its value is the empty set. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- (-. Fun (F |` {A}) -> (F` A) = (/))
Theorem	nfunsnOLD 4707	If the restriction of a class to a singleton is not a function, its value is the empty set.
|- (-. Fun (F |` {A}) -> (F` A) = (/))
Theorem	fveqres 4708	Equal values imply equal values in a restriction.
|- ((F` A) = (G` A) -> ((F |` B)` A) = ((G |` B)` A))
Theorem	funbrfv 4709	The second argument of a binary relation on a function is the function's value.
|- B e. _V   =>   |- (Fun F -> (AFB -> (F` A) = B))
Theorem	funopfv 4710	The second element in an ordered pair member of a function is the function's value.
|- B e. _V   =>   |- (Fun F -> (<.A, B>. e. F -> (F` A) = B))
Theorem	funopfvg 4711	The second element in an ordered pair member of a function is the function's value.
|- ((B e. C /\ Fun F) -> (<.A, B>. e. F -> (F` A) = B))
Theorem	fnbrfvb 4712	Equivalence of function value and binary relation.
|- C e. _V   =>   |- ((F Fn A /\ B e. A) -> ((F` B) = C <-> BFC))
Theorem	fnopfvb 4713	Equivalence of function value and ordered pair membership.
|- C e. _V   =>   |- ((F Fn A /\ B e. A) -> ((F` B) = C <-> <.B, C>. e. F))
Theorem	funbrfvb 4714	Equivalence of function value and binary relation.
|- B e. _V   =>   |- ((Fun F /\ A e. dom F) -> ((F` A) = B <-> AFB))
Theorem	funopfvb 4715	Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42.
|- B e. _V   =>   |- ((Fun F /\ A e. dom F) -> ((F` A) = B <-> <.A, B>. e. F))
Theorem	funbrfvbg 4716	Function value in terms of a binary relation.
|- ((Fun F /\ A e. dom F /\ B e. C) -> ((F` A) = B <-> AFB))
Theorem	dffn5 4717	Representation of a function in terms of its values.
|- (F Fn A <-> F = {<.x, y>. | (x e. A /\ y = (F` x))})
Theorem	fnrnfv 4718	The range of a function expressed as a collection of the function's values.
|- (F Fn A -> ran F = {y | E.x e. A y = (F` x)})
Theorem	fvelrnb 4719	A member of a function's range is a value of the function.
|- (F Fn A -> (B e. ran F <-> E.x e. A (F` x) = B))
Theorem	dfimafn 4720	Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)
|- ((Fun F /\ A C_ dom F) -> (F"A) = {y | E.x e. A (F` x) = y})
Theorem	dfimafn2 4721	Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)
|- ((Fun F /\ A C_ dom F) -> (F"A) = U_x e. A {(F` x)})
Theorem	funimass4 4722	Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)
|- ((Fun F /\ A C_ dom F) -> ((F"A) C_ B <-> A.x e. A (F` x) e. B))
Theorem	fvelima 4723	Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- ((Fun F /\ A e. (F"B)) -> E.x e. B (F` x) = A)
Theorem	fvelimaOLD 4724	Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42.
|- ((Fun F /\ A e. (F"B)) -> E.x e. B (F` x) = A)
Theorem	fvelimab 4725	Function value in an image. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (An unnecessary distinct variable restriction was removed by David Abernethy, 17-Dec-2011.)
|- ((F Fn A /\ B C_ A) -> (C e. (F"B) <-> E.x e. B (F` x) = C))
Theorem	fvelimabOLD 4726	Function value in an image.
|- ((F Fn A /\ B C_ A) -> (C e. (F"B) <-> E.x e. B (F` x) = C))
Theorem	fniinfv 4727	The indexed intersection of a function's values is the intersection of its range.
|- (F Fn A -> |^|_x e. A (F` x) = |^|ran F)
Theorem	fnsnfv 4728	Singleton of function value.
|- ((F Fn A /\ B e. A) -> {(F` B)} = (F"{B}))
Theorem	ssimaex 4729	The existence of a subimage.
|- A e. _V   =>   |- ((Fun F /\ B C_ (F"A)) -> E.x(x C_ A /\ B = (F"x)))
Theorem	ssimaexg 4730	The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
|- ((A e. C /\ Fun F /\ B C_ (F"A)) -> E.x(x C_ A /\ B = (F"x)))
Theorem	funfv 4731	A simplified expression for the value of a function when we know it's a function.
|- (Fun F -> (F` A) = U.(F"{A}))
Theorem	funfv2 4732	The value of a function. Definition of function value in [Enderton] p. 43.
|- (Fun F -> (F` A) = U.{y | AFy})
Theorem	funfv2f 4733	The value of a function. Version of funfv2 4732 using a bound-variable hypotheses instead of distinct variable conditions.
|- (z e. A -> A.y z e. A)   &   |- (z e. F -> A.y z e. F)   =>   |- (Fun F -> (F` A) = U.{y | AFy})
Theorem	dffv2 4734	Alternate definition of function value df-fv 4014 that doesn't require dummy variables.
|- (F` A) = U.((F"{A}) \ U.U.(((F |` {A}) o. `'(F |` {A})) \ _I ))
Theorem	dmfco 4735	Domains of a function composition.
|- ((Fun G /\ A e. dom G) -> (A e. dom ( F o. G) <-> (G` A) e. dom F))
Theorem	fvco 4736	Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28.
|- ((Fun F /\ Fun G /\ A e. dom G) -> ((F o. G)` A) = (F` (G` A)))
Theorem	fvco2 4737	Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- ((Fun F /\ G Fn A /\ C e. A) -> ((F o. G)` C) = (F` (G` C)))
Theorem	fvco2OLD 4738	Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47.
|- ((Fun F /\ G Fn A /\ C e. A) -> ((F o. G)` C) = (F` (G` C)))
Theorem	fvco3 4739	Value of a function composition.
|- ((Fun F /\ G:A-->B /\ C e. A) -> ((F o. G)` C) = (F` (G` C)))
Theorem	fvopab3 4740	Value of a function given by ordered-pair class abstraction.
|- B e. _V   &   |- (x = A -> (ph <-> ps))   &   |- (y = B -> (ps <-> ch))   &   |- (x e. C -> E!yph)   &   |- F = {<.x, y>. | (x e. C /\ ph)}   =>   |- (A e. C -> ((F` A) = B <-> ch))
Theorem	fvopab3ig 4741	Value of a function given by ordered-pair class abstraction.
|- (x = A -> (ph <-> ps))   &   |- (y = B -> (ps <-> ch))   &   |- (x e. C -> E*yph)   &   |- F = {<.x, y>. | (x e. C /\ ph)}   =>   |- ((A e. C /\ B e. D) -> (ch -> (F` A) = B))
Theorem	fvopab4g 4742	Value of a function given by ordered-pair class abstraction.
|- (x = A -> B = C)   &   |- F = {<.x, y>. | (x e. D /\ y = B)}   =>   |- ((A e. D /\ C e. R) -> (F` A) = C)
Theorem	fvopab4 4743	Value of a function given by ordered-pair class abstraction.
|- (x = A -> B = C)   &   |- F = {<.x, y>. | (x e. D /\ y = B)}   &   |- C e. _V   =>   |- (A e. D -> (F` A) = C)
Theorem	fvopab4gf 4744	Value of a function given by an ordered-pair class abstraction. This version of fvopab4g 4742 uses bound-variable hypotheses instead of distinct variable conditions.
|- (z e. A -> A.x z e. A)   &   |- (z e. C -> A.x z e. C)   &   |- (x = A -> B = C)   &   |- F = {<.x, y>. | (x e. D /\ y = B)}   =>   |- ((A e. D /\ C e. R) -> (F` A) = C)
Theorem	fvopab4sf 4745	Value of a function given by ordered-pair class abstraction, using explicit class substitution.
|- A e. _V   &   |- B e. _V   &   |- (z e. A -> A.x z e. A)   &   |- F = {<.x, y>. | (x e. C /\ y = B)}   =>   |- (A e. C -> (F` A) = [_A / x]_B)
Theorem	fvopab4s 4746	Value of a function given by ordered-pair class abstraction, using explicit class substitution.
|- A e. _V   &   |- B e. _V   &   |- F = {<.x, y>. | (x e. C /\ y = B)}   =>   |- (A e. C -> (F` A) = [_A / x]_B)
Theorem	fvopab4ndm 4747	Value of a function given by an ordered-pair class abstraction, outside of its domain.
|- F = {<.x, y>. | (x e. A /\ ph)}   =>   |- (-. B e. A -> (F` B) = (/))
Theorem	fvopabg 4748	The value of a function given by ordered-pair class abstraction.
|- (x = A -> B = C)   =>   |- ((A e. D /\ C e. R) -> ({<.x, y>. | y = B}` A) = C)
Theorem	fvopabn 4749	This somewhat non-intuitive theorem tells us the value of its function is the empty set when the class C it would otherwise map to is a proper class. This is a technical lemma that can help eliminate redundant sethood antecedents otherwise required by fvopabg 4748.
|- (x = A -> B = C)   =>   |- (-. C e. _V -> ({<.x, y>. | y = B}` A) = (/))
Theorem	fvopabgf 4750	The value of a function given by ordered-pair class abstraction.
|- (z e. A -> A.x z e. A)   &   |- (z e. C -> A.x z e. C)   &   |- (x = A -> B = C)   =>   |- ((A e. D /\ C e. R) -> ({<.x, y>. | y = B}` A) = C)
Theorem	fvopabnf 4751	The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvopabn 4749 uses bound-variable hypotheses instead of distinct variable conditions.
|- (z e. A -> A.x z e. A)   &   |- (z e. C -> A.x z e. C)   &   |- (x = A -> B = C)   =>   |- (-. C e. _V -> ({<.x, y>. | y = B}` A) = (/))
Theorem	fvopabf 4752	The value of a function given by ordered-pair class abstraction.
|- (z e. A -> A.x z e. A)   &   |- (z e. C -> A.x z e. C)   &   |- A e. _V   &   |- C e. _V   &   |- (x = A -> B = C)   =>   |- ({<.x, y>. | y = B}` A) = C
Theorem	fvopab 4753	The value of a function given by an ordered-pair class abstraction.
|- A e. _V   &   |- C e. _V   &   |- (x = A -> B = C)   =>   |- ({<.x, y>. | y = B}` A) = C
Theorem	fvopab2 4754	Value of a function given by an ordered-pair class abstraction.
|- ((x e. A /\ B e. C) -> ({<.x, y>. | (x e. A /\ y = B)}` x) = B)
Theorem	fvopabs 4755	The value of a function given by an ordered-pair class abstraction, using class substitution.
|- A e. _V   &   |- B e. _V   =>   |- ({<.x, y>. | y = B}` A) = [_A / x]_B
Theorem	fvopab5 4756	The value of a function that is expressed as an ordered pair abstraction.
|- F = {<.x, y>. | ph}   &   |- (x = A -> (ph <-> ps))   =>   |- ((Fun F /\ A e. B) -> (F` A) = U.{y | ps})
Theorem	fvopab6 4757	Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- F = {<.x, y>. | (ph /\ y = B)}   &   |- (x = A -> (ph <-> ps))   &   |- (x = A -> B = C)   =>   |- ((A e. D /\ C e. R /\ ps) -> (F` A) = C)
Theorem	fvsn 4758	The value of a singleton of an ordered pair is the second member.
|- A e. _V   &   |- B e. _V   =>   |- ({<.A, B>.}` A) = B
Theorem	fvpr1 4759	The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
|- A e. _V   &   |- B e. _V   &   |- C e. _V   &   |- D e. _V   =>   |- (A =/= B -> ({<.A, C>., <.B, D>.}` A) = C)
Theorem	fvpr2 4760	The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
|- A e. _V   &   |- B e. _V   &   |- C e. _V   &   |- D e. _V   =>   |- (A =/= B -> ({<.A, C>., <.B, D>.}` B) = D)
Theorem	fvtp1 4761	The first value of a function with a domain of three elements.
|- A e. _V   &   |- B e. _V   &   |- C e. _V   &   |- D e. _V   &   |- E e. _V   &   |- F e. _V   =>   |- ((A =/= B /\ A =/= C /\ B =/= C) -> ({<.A, D>., <.B, E>., <.C, F>.}` A) = D)
Theorem	fvtp2 4762	The second value of a function with a domain of three elements.
|- A e. _V   &   |- B e. _V   &   |- C e. _V   &   |- D e. _V   &   |- E e. _V   &   |- F e. _V   =>   |- ((A =/= B /\ A =/= C /\ B =/= C) -> ({<.A, D>., <.B, E>., <.C, F>.}` B) = E)
Theorem	fvtp3 4763	The third value of a function with a domain of three elements.
|- A e. _V   &   |- B e. _V   &   |- C e. _V   &   |- D e. _V   &   |- E e. _V   &   |- F e. _V   =>   |- ((A =/= B /\ A =/= C /\ B =/= C) -> ({<.A, D>., <.B, E>., <.C, F>.}` C) = F)
Theorem	fvsnun1 4764	The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 4765.
|- A e. _V   &   |- B e. _V   &   |- G = ({<.A, B>.} u. (F |` (C \ {A})))   =>   |- (G` A) = B
Theorem	fvsnun2 4765	The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 4764.
|- A e. _V   &   |- B e. _V   &   |- G = ({<.A, B>.} u. (F |` (C \ {A})))   =>   |- (D e. (C \ {A}) -> (G` D) = (F` D))
Theorem	eqfnfv 4766	Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28.
|- ((F Fn A /\ G Fn B) -> (F = G <-> (A = B /\ A.x e. A (F` x) = (G` x))))
Theorem	eqfnfv2 4767	Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- ((F Fn A /\ G Fn A) -> (F = G <-> A.x e. A (F` x) = (G` x)))
Theorem	eqfnfv2OLD 4768	Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted).
|- ((F Fn A /\ G Fn A) -> (F = G <-> A.x e. A (F` x) = (G` x)))
Theorem	eqfnfv3 4769	Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ((F Fn A /\ G Fn B) -> (F = G <-> (B C_ A /\ A.x e. A (x e. B /\ (F` x) = (G` x)))))
Theorem	eqfnfv2f 4770	Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv2 4767 uses bound-variable hypotheses instead of distinct variable conditions.
|- (y e. F -> A.x y e. F)   &   |- (y e. G -> A.x y e. G)   =>   |- ((F Fn A /\ G Fn A) -> (F = G <-> A.x e. A (F` x) = (G` x)))
Theorem	eufnfv 4771	A function is uniquely determined by its values.
|- A e. _V   &   |- B e. _V   =>   |- E!f(f Fn A /\ A.x e. A (f` x) = B)
Theorem	fvreseq 4772	Equality of restricted functions is determined by their values.
|- (((F Fn A /\ G Fn A) /\ B C_ A) -> ((F |` B) = (G |` B) <-> A.x e. B (F` x) = (G` x)))
Theorem	elrnopabg 4773	Membership in the range of an ordered pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = B)}   =>   |- (A.x e. A B e. D -> (C e. ran F <-> E.x e. A C = B))
Theorem	elrnopab 4774	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 4775	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 C_ U.A)
Theorem	funfvop 4776	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 4777	A member of a preimage is a function value argument.
|- ((Fun F /\ A e. (`'F"B)) -> (F` A) e. B)
Theorem	fvimacnv 4778	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 4492 could probably be strengthened to a biconditional.")
|- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> A e. (`'F"B)))
Theorem	funimass3 4779	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 4778 would be the special case of A being a singleton, but it works this way round too.")
|- ((Fun F /\ A C_ dom F) -> ((F"A) C_ B <-> A C_ (`'F"B)))
Theorem	funimass5 4780	A subclass of a preimage in terms of function values.
|- ((Fun F /\ A C_ dom F) -> (A C_ (`'F"B) <-> A.x e. A (F` x) e. B))
Theorem	funconstss 4781	Two ways of specifying that a function is constant on a subdomain.
|- ((Fun F /\ A C_ dom F) -> (A.x e. A (F` x) = B <-> A C_ (`'F"{B})))
Theorem	fvimacnvALT 4782	Another proof of fvimacnv 4778, based on funimass3 4779. If funimass3 4779 is ever proved directly, as opposed to using funimacnv 4490 pointwise, then the proof of funimacnv 4490 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 4783	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 4784	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 4785	A function's value belongs to its range.
|- ((Fun F /\ A e. dom F) -> (F` A) e. ran F)
Theorem	fnfvelrn 4786	A function's value belongs to its range.
|- ((F Fn A /\ B e. A) -> (F` B) e. ran F)
Theorem	ffvelrn 4787	A function's value belongs to its codomain.
|- ((F:A-->B /\ C e. A) -> (F` C) e. B)
Theorem	ffvelrni 4788	A function's value belongs to its codomain.
|- F:A-->B   =>   |- (C e. A -> (F` C) e. B)
Theorem	dff2 4789	Alternate definition of a mapping.
|- (F:A-->B <-> (F Fn A /\ F C_ (A X. B)))
Theorem	dff3 4790	Alternate definition of a mapping.
|- (F:A-->B <-> (F C_ (A X. B) /\ A.x e. A E!y xFy))
Theorem	dff4 4791	Alternate definition of a mapping.
|- (F:A-->B <-> (F C_ (A X. B) /\ A.x e. A E!y e. B xFy))
Theorem	dffo3 4792	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 4793	Alternate definition of an onto mapping.
|- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x e. A xFy))
Theorem	dffo5 4794	Alternate definition of an onto mapping.
|- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x xFy))
Theorem	exfo 4795	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 4796	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 4797	Inclusion of a function in a cross product.
|- F = {<.x, y>. | (x e. A /\ y = C)}   =>   |- (A.x e. A C e. B -> F C_ (A X. B))
Theorem	rnssopab 4798	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 C_ B)
Theorem	fopab3 4799	Functionality of an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = C)}   &   |- C e. _V   =>   |- (ran F C_ B <-> F:A-->B)
Theorem	fopab 4800	Functionality of an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = C)}   &   |- (x e. A -> C e. B)   =>   |- F:A-->B