mmtheorems47 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4601-4700 - Page 47 of 175

Type	Label	Description
Statement
Theorem	f00 4601	A class is a function with empty codomain iff it and its domain are empty.
|- (F:A-->(/) <-> (F = (/) /\ A = (/)))
Theorem	fconst 4602	A cross product with a singleton is a constant function. (The proof was shortened by Andrew Salmon, 17-Sep-2011.)
|- B e. _V   =>   |- (A X. {B}):A-->{B}
Theorem	fconstOLD 4603	A cross product with a singleton is a constant function.
|- B e. _V   =>   |- (A X. {B}):A-->{B}
Theorem	fconstg 4604	A cross product with a singleton is a constant function.
|- (B e. C -> (A X. {B}):A-->{B})
Theorem	f1eq1 4605	Equality theorem for one-to-one functions.
|- (F = G -> (F:A-1-1->B <-> G:A-1-1->B))
Theorem	f1eq2 4606	Equality theorem for one-to-one functions.
|- (A = B -> (F:A-1-1->C <-> F:B-1-1->C))
Theorem	f1eq3 4607	Equality theorem for one-to-one functions.
|- (A = B -> (F:C-1-1->A <-> F:C-1-1->B))
Theorem	hbf1 4608	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 4609	Alternate definition of a one-to-one function.
|- (F:A-1-1->B <-> (F:A-->B /\ A.yE*x xFy))
Theorem	f1f 4610	A one-to-one mapping is a mapping.
|- (F:A-1-1->B -> F:A-->B)
Theorem	f1cnv 4611	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 4612	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 4613	Equality theorem for onto functions.
|- (F = G -> (F:A-onto->B <-> G:A-onto->B))
Theorem	foeq2 4614	Equality theorem for onto functions.
|- (A = B -> (F:A-onto->C <-> F:B-onto->C))
Theorem	foeq3 4615	Equality theorem for onto functions.
|- (A = B -> (F:C-onto->A <-> F:C-onto->B))
Theorem	hbfo 4616	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 4617	An onto mapping is a mapping.
|- (F:A-onto->B -> F:A-->B)
Theorem	fofun 4618	An onto mapping is a function.
|- (F:A-onto->B -> Fun F)
Theorem	fofn 4619	An onto mapping is a function on its domain.
|- (F:A-onto->B -> F Fn A)
Theorem	forn 4620	The codomain of an onto function is its range.
|- (F:A-onto->B -> ran F = B)
Theorem	dffo2 4621	Alternate definition of an onto function.
|- (F:A-onto->B <-> (F:A-->B /\ ran F = B))
Theorem	foima 4622	The image of the domain of an onto function.
|- (F:A-onto->B -> (F"A) = B)
Theorem	dffn4 4623	A function maps onto its range.
|- (F Fn A <-> F:A-onto->ran F)
Theorem	funforn 4624	A function maps its domain onto its range.
|- (Fun A <-> A:dom A-onto->ran A)
Theorem	fornex 4625	If the domain of an onto function exists, so does its codomain.
|- (A e. C -> (F:A-onto->B -> B e. _V))
Theorem	fodmrnu 4626	An onto function has unique domain and range.
|- ((F:A-onto->B /\ F:C-onto->D) -> (A = C /\ B = D))
Theorem	fores 4627	Restriction of a function.
|- ((Fun F /\ A C_ dom F) -> (F |` A):A-onto->(F"A))
Theorem	foco 4628	Composition of onto functions.
|- ((F:B-onto->C /\ G:A-onto->B) -> (F o. G):A-onto->C)
Theorem	foconst 4629	A nonzero constant function is onto.
|- ((F:A-->{B} /\ F =/= (/)) -> F:A-onto->{B})
Theorem	f1oeq1 4630	Equality theorem for one-to-one onto functions.
|- (F = G -> (F:A-1-1-onto->B <-> G:A-1-1-onto->B))
Theorem	f1oeq2 4631	Equality theorem for one-to-one onto functions.
|- (A = B -> (F:A-1-1-onto->C <-> F:B-1-1-onto->C))
Theorem	f1oeq3 4632	Equality theorem for one-to-one onto functions.
|- (A = B -> (F:C-1-1-onto->A <-> F:C-1-1-onto->B))
Theorem	hbf1o 4633	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 4634	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 4635	A one-to-one onto mapping is a mapping.
|- (F:A-1-1-onto->B -> F:A-->B)
Theorem	f1ofn 4636	A one-to-one onto mapping is function on its domain.
|- (F:A-1-1-onto->B -> F Fn A)
Theorem	f1ofun 4637	A one-to-one onto mapping is a function.
|- (F:A-1-1-onto->B -> Fun F)
Theorem	f1orel 4638	A one-to-one onto mapping is a relation.
|- (F:A-1-1-onto->B -> Rel F)
Theorem	dff1o2 4639	Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))
Theorem	dff1o2OLD 4640	Alternate definition of one-to-one onto function.
|- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))
Theorem	dff1o3 4641	Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- (F:A-1-1-onto->B <-> (F:A-onto->B /\ Fun `'F))
Theorem	dff1o3OLD 4642	Alternate definition of one-to-one onto function.
|- (F:A-1-1-onto->B <-> (F:A-onto->B /\ Fun `'F))
Theorem	f1ofo 4643	A one-to-one onto function is an onto function.
|- (F:A-1-1-onto->B -> F:A-onto->B)
Theorem	dff1o4 4644	Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- (F:A-1-1-onto->B <-> (F Fn A /\ `'F Fn B))
Theorem	dff1o4OLD 4645	Alternate definition of one-to-one onto function.
|- (F:A-1-1-onto->B <-> (F Fn A /\ `'F Fn B))
Theorem	dff1o5 4646	Alternate definition of one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ ran F = B))
Theorem	dff1o5OLD 4647	Alternate definition of one-to-one onto function.
|- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ ran F = B))
Theorem	f1orn 4648	A one-to-one function maps onto its range.
|- (F:A-1-1-onto->ran F <-> (F Fn A /\ Fun `'F))
Theorem	f1f1orn 4649	A one-to-one function maps one-to-one onto its range.
|- (F:A-1-1->B -> F:A-1-1-onto->ran F)
Theorem	f1oabexg 4650	The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)
|- F = {f | (f:A-1-1-onto->B /\ ph)}   =>   |- ((A e. C /\ B e. D) -> F e. _V)
Theorem	f1ocnv 4651	The converse of a one-to-one onto function is also one-to-one onto. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)
Theorem	f1ocnvOLD 4652	The converse of a one-to-one onto function is also one-to-one onto.
|- (F:A-1-1-onto->B -> `'F:B-1-1-onto->A)
Theorem	f1ocnvb 4653	A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged.
|- (Rel F -> (F:A-1-1-onto->B <-> `'F:B-1-1-onto->A))
Theorem	f1ores 4654	The restriction of a one-to-one function maps one-to-one onto the image.
|- ((F:A-1-1->B /\ C C_ A) -> (F |` C):C-1-1-onto->(F"C))
Theorem	f1orescnv 4655	The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)
|- ((Fun `'F /\ (F |` R):R-1-1-onto->P) -> (`'F |` P):P-1-1-onto->R)
Theorem	f1imacnv 4656	Pre-image of an image.
|- ((F:A-1-1->B /\ C C_ A) -> (`'F"(F"C)) = C)
Theorem	foimacnv 4657	A reverse version of f1imacnv 4656. (Contributed by Jeffrey Hankins, 16-Jul-2009.)
|- ((F:A-onto->B /\ C C_ B) -> (F"(`'F"C)) = C)
Theorem	f1oun 4658	The union of two one-to-one onto functions with disjoint domains and ranges.
|- (((F:A-1-1-onto->B /\ G:C-1-1-onto->D) /\ ((A i^i C) = (/) /\ (B i^i D) = (/))) -> (F u. G):(A u. C)-1-1-onto->(B u. D))
Theorem	resdif 4659	The restriction of a one-to-one onto function to a difference maps onto the difference of the images. (Contributed by Paul Chapman, 11-Apr-2009.)
|- ((Fun `'F /\ (F |` A):A-onto->C /\ (F |` B):B-onto->D) -> (F |` (A \ B)):(A \ B)-1-1-onto->(C \ D))
Theorem	resin 4660	The restriction of a one-to-one onto function to an intersection maps onto the intersection of the images. (Contributed by Paul Chapman, 11-Apr-2009.)
|- ((Fun `'F /\ (F |` A):A-onto->C /\ (F |` B):B-onto->D) -> (F |` (A i^i B)):(A i^i B)-1-1-onto->(C i^i D))
Theorem	f1oco 4661	Composition of one-to-one onto functions.
|- ((F:B-1-1-onto->C /\ G:A-1-1-onto->B) -> (F o. G):A-1-1-onto->C)
Theorem	f1ococnv2 4662	The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range.
|- (F:A-1-1-onto->B -> (F o. `'F) = ( _I |` B))
Theorem	f1ococnv1 4663	The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain.
|- (F:A-1-1-onto->B -> (`'F o. F) = ( _I |` A))
Theorem	f1dmex 4664	If the codomain of a one-to-one function exists, so does its domain. This theorem is equivalent to the Axiom of Replacement ax-rep 3428.
|- ((F:A-1-1->B /\ B e. C) -> A e. _V)
Theorem	ffoss 4665	Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145.
|- F e. _V   =>   |- (F:A-->B <-> E.x(F:A-onto->x /\ x C_ B))
Theorem	f11o 4666	Relationship between one-to-one and one-to-one onto function.
|- F e. _V   =>   |- (F:A-1-1->B <-> E.x(F:A-1-1-onto->x /\ x C_ B))
Theorem	f10 4667	The empty set maps one-to-one into any class.
|- (/):(/)-1-1->A
Theorem	f1o00 4668	One-to-one onto mapping of the empty set.
|- (F:(/)-1-1-onto->A <-> (F = (/) /\ A = (/)))
Theorem	fo00 4669	Onto mapping of the empty set.
|- (F:(/)-onto->A <-> (F = (/) /\ A = (/)))
Theorem	f1o0 4670	One-to-one onto mapping of the empty set.
|- (/):(/)-1-1-onto->(/)
Theorem	f1oi 4671	A restriction of the identity relation is a one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- ( _I |` A):A-1-1-onto->A
Theorem	f1oiOLD 4672	A restriction of the identity relation is a one-to-one onto function.
|- ( _I |` A):A-1-1-onto->A
Theorem	f1ovi 4673	The identity relation is a one-to-one onto function on the universe.
|- _I :_V-1-1-onto->_V
Theorem	f1osn 4674	A singleton of an ordered pair is one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)
|- A e. _V   &   |- B e. _V   =>   |- {<.A, B>.}:{A}-1-1-onto->{B}
Theorem	f1osnOLD 4675	A singleton of an ordered pair is one-to-one onto function.
|- A e. _V   &   |- B e. _V   =>   |- {<.A, B>.}:{A}-1-1-onto->{B}
Theorem	fv2 4676	Alternate definition of function value. Definition 10.11 of [Quine] p. 68. (The proof was shortened by Andrew Salmon, 17-Sep-2011.)
|- A e. _V   =>   |- (F` A) = U.{x | A.y(AFy <-> y = x)}
Theorem	fv2OLD 4677	Alternate definition of function value. Definition 10.11 of [Quine] p. 68.
|- A e. _V   =>   |- (F` A) = U.{x | A.y(AFy <-> y = x)}
Theorem	fvprc 4678	A function's value at a proper class is the empty set.
|- (-. A e. _V -> (F` A) = (/))
Theorem	elfv 4679	Membership in a function value.
|- B e. _V   =>   |- (A e. (F` B) <-> E.x(A e. x /\ A.y(BFy <-> y = x)))
Theorem	fveq1 4680	Equality theorem for function value.
|- (F = G -> (F` A) = (G` A))
Theorem	fveq2 4681	Equality theorem for function value.
|- (A = B -> (F` A) = (F` B))
Theorem	fveq1i 4682	Equality inference for function value.
|- F = G   =>   |- (F` A) = (G` A)
Theorem	fveq1d 4683	Equality deduction for function value.
|- (ph -> F = G)   =>   |- (ph -> (F` A) = (G` A))
Theorem	fveq2i 4684	Equality inference for function value.
|- A = B   =>   |- (F` A) = (F` B)
Theorem	fveq2d 4685	Equality deduction for function value.
|- (ph -> A = B)   =>   |- (ph -> (F` A) = (F` B))
Theorem	hbfv 4686	Bound-variable hypothesis builder for function value.
|- (y e. F -> A.x y e. F)   &   |- (y e. A -> A.x y e. A)   =>   |- (y e. (F` A) -> A.x y e. (F` A))
Theorem	hbfvd 4687	Deduction version of bound-variable hypothesis builder hbfv 4686. If a closed theorem version is desired, see hbfvd2 4688.
|- (ph -> A.xph)   &   |- (ph -> (y e. F -> A.x y e. F))   &   |- (ph -> (y e. A -> A.x y e. A))   =>   |- (ph -> (y e. (F` A) -> A.x y e. (F` A)))
Theorem	hbfvd2 4688	Deduction version of bound-variable hypothesis builder hbfv 4686. This variant of hbfvd 4687 allows us to create a closed theorem form by replacing the uncommitted antecedent ph with an appropriate substitution instance.
|- (ph -> A.xA.yph)   &   |- (ph -> (y e. F -> A.x y e. F))   &   |- (ph -> (y e. A -> A.x y e. A))   =>   |- (ph -> (y e. (F` A) -> A.x y e. (F` A)))
Theorem	fvex 4689	The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27.
|- (F` A) e. _V
Theorem	fv3 4690	Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26.
|- A e. _V   =>   |- (F` A) = {x | (E.y(x e. y /\ AFy) /\ E!y AFy)}
Theorem	fvres 4691	The value of a restricted function.
|- (A e. B -> ((F |` B)` A) = (F` A))
Theorem	funssfv 4692	The value of a member of the domain of a subclass of a function.
|- ((Fun F /\ G C_ F /\ A e. dom G) -> (F` A) = (G` A))
Theorem	tz6.12-1 4693	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27.
|- A e. _V   =>   |- ((AFy /\ E!y AFy) -> (F` A) = y)
Theorem	tz6.12 4694	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27.
|- A e. _V   =>   |- ((<.A, y>. e. F /\ E!y<.A, y>. e. F) -> (F` A) = y)
Theorem	tz6.12f 4695	Function value, using bound-variable hypotheses instead of distinct variable conditions.
|- (w e. F -> A.y w e. F)   =>   |- ((<.x, y>. e. F /\ E!y<.x, y>. e. F) -> (F` x) = y)
Theorem	tz6.12-2 4696	Function value when F is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27.
|- (-. E!y AFy -> (F` A) = (/))
Theorem	tz6.12c 4697	Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27.
|- A e. _V   =>   |- (E!y AFy -> ((F` A) = y <-> AFy))
Theorem	tz6.12i 4698	Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27.
|- A e. _V   =>   |- (B =/= (/) -> ((F` A) = B -> AFB))
Theorem	csbfv12g 4699	Move class substitution in and out of a function value.
|- (A e. C -> [_A / x]_(F` B) = ([_A / x]_F` [_A / x]_B))
Theorem	csbfv2g 4700	Move class substitution in and out of a function value.
|- (A e. C -> [_A / x]_(F` B) = (F` [_A / x]_B))