mmtheorems39 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3801-3900 - Page 39 of 123

Type	Label	Description
Statement
Theorem	dff1o2 3801	Alternate definition of one-to-one onto function.
|- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))
Theorem	dff1o3 3802	Alternate definition of one-to-one onto function.
|- (F:A-1-1-onto->B <-> (F:A-onto->B /\ Fun `'F))
Theorem	f1ofo 3803	A one-to-one onto function is an onto function.
|- (F:A-1-1-onto->B -> F:A-onto->B)
Theorem	dff1o4 3804	Alternate definition of one-to-one onto function.
|- (F:A-1-1-onto->B <-> (F Fn A /\ `'F Fn B))
Theorem	dff1o5 3805	Alternate definition of one-to-one onto function.
|- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ ran F = B))
Theorem	f1orn 3806	A one-to-one function maps onto its range.
|- (F:A-1-1-onto->ran F <-> (F Fn A /\ Fun `'F))
Theorem	f1f1orn 3807	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 3808	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 3809	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 3810	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 3811	The restriction of a one-to-one function maps one-to-one onto the image.
|- ((F:A-1-1->B /\ C (_ A) -> (F |` C):C-1-1-onto->(F"C))
Theorem	f1orescnv 3812	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 3813	Pre-image of an image.
|- ((F:A-1-1->B /\ C (_ A) -> (`'F"(F"C)) = C)
Theorem	foimacnv 3814	A reverse version of f1imacnv 3813. (Contributed by Jeffrey Hankins, 16-Jul-2009.)
|- ((F:A-onto->B /\ C (_ B) -> (F"(`'F"C)) = C)
Theorem	f1oun 3815	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 3816	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 3817	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 3818	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 3819	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 3820	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 3821	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 2767.
|- ((F:A-1-1->B /\ B e. C) -> A e. V)
Theorem	ffoss 3822	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 (_ B))
Theorem	f11o 3823	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 (_ B))
Theorem	f10 3824	The empty set maps one-to-one into any class.
|- (/):(/)-1-1->A
Theorem	f1o00 3825	One-to-one onto mapping of the empty set.
|- (F:(/)-1-1-onto->A <-> (F = (/) /\ A = (/)))
Theorem	fo00 3826	Onto mapping of the empty set.
|- (F:(/)-onto->A <-> (F = (/) /\ A = (/)))
Theorem	f1o0 3827	One-to-one onto mapping of the empty set.
|- (/):(/)-1-1-onto->(/)
Theorem	f1oi 3828	A restriction of the identity relation is a one-to-one onto function.
|- (I |` A):A-1-1-onto->A
Theorem	f1ovi 3829	The identity relation is a one-to-one onto function on the universe.
|- I:V-1-1-onto->V
Theorem	f1osn 3830	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 3831	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 3832	A function's value at a proper class is the empty set.
|- (-. A e. V -> (F` A) = (/))
Theorem	elfv 3833	Membership in a function value.
|- B e. V   =>   |- (A e. (F` B) <-> E.x(A e. x /\ A.y(BFy <-> y = x)))
Theorem	fveq1 3834	Equality theorem for function value.
|- (F = G -> (F` A) = (G` A))
Theorem	fveq2 3835	Equality theorem for function value.
|- (A = B -> (F` A) = (F` B))
Theorem	fveq1i 3836	Equality inference for function value.
|- F = G   =>   |- (F` A) = (G` A)
Theorem	fveq1d 3837	Equality deduction for function value.
|- (ph -> F = G)   =>   |- (ph -> (F` A) = (G` A))
Theorem	fveq2i 3838	Equality inference for function value.
|- A = B   =>   |- (F` A) = (F` B)
Theorem	fveq2d 3839	Equality deduction for function value.
|- (ph -> A = B)   =>   |- (ph -> (F` A) = (F` B))
Theorem	hbfv 3840	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 3841	Deduction version of bound-variable hypothesis builder hbfv 3840. If a closed theorem version is desired, see hbfvd2 3842.
|- (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 3842	Deduction version of bound-variable hypothesis builder hbfv 3840. This variant of hbfvd 3841 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 3843	The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27.
|- (F` A) e. V
Theorem	fv3 3844	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 3845	The value of a restricted function.
|- (A e. B -> ((F |` B)` A) = (F` A))
Theorem	funssfv 3846	The value of a member of the domain of a subclass of a function.
|- ((Fun F /\ G (_ F /\ A e. dom G) -> (F` A) = (G` A))
Theorem	tz6.12-1 3847	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27.
|- A e. V   =>   |- ((AFy /\ E!y AFy) -> (F` A) = y)
Theorem	tz6.12 3848	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 3849	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 3850	Function value when F is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27.
|- (-. E!y AFy -> (F` A) = (/))
Theorem	tz6.12c 3851	Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27.
|- A e. V   =>   |- (E!y AFy -> ((F` A) = y <-> AFy))
Theorem	tz6.12i 3852	Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27.
|- A e. V   =>   |- (B =/= (/) -> ((F` A) = B -> AFB))
Theorem	csbfv12g 3853	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 3854	Move class substitution in and out of a function value.
|- (A e. C -> [_A / x]_(F` B) = (F` [_A / x]_B))
Theorem	csbfvg 3855	Substitution for a function value.
|- (A e. C -> [_A / x]_(F` x) = (F` A))
Theorem	ndmfv 3856	The value of a class outside its domain is the empty set.
|- (-. A e. dom F -> (F` A) = (/))
Theorem	ndmfvrcl 3857	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 3858	If a function value has a member, the argument belongs to the domain.
|- (A e. (F` B) -> B e. dom F)
Theorem	nfvres 3859	A non-element of a restriction has empty value.
|- (-. A e. B -> ((F |` B)` A) = (/))
Theorem	fveqres 3860	Equal values imply equal values in a restriction.
|- ((F` A) = (G` A) -> ((F |` B)` A) = ((G |` B)` A))
Theorem	funbrfv 3861	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 3862	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 3863	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 3864	Equivalence of function value and binary relation.
|- C e. V   =>   |- ((F Fn A /\ B e. A) -> ((F` B) = C <-> BFC))
Theorem	fnopfvb 3865	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 3866	Equivalence of function value and binary relation.
|- B e. V   =>   |- ((Fun F /\ A e. dom F) -> ((F` A) = B <-> AFB))
Theorem	funopfvb 3867	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 3868	Function value in terms of a binary relation.
|- ((Fun F /\ A e. dom F /\ B e. C) -> ((F` A) = B <-> AFB))
Theorem	dffn5 3869	Representation of a function in terms of its values.
|- (F Fn A <-> F = {<.x, y>. | (x e. A /\ y = (F` x))})
Theorem	fnrnfv 3870	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 3871	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 3872	Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)
|- ((Fun F /\ A (_ dom F) -> (F"A) = {y | E.x e. A (F` x) = y})
Theorem	dfimafn2 3873	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 (_ dom F) -> (F"A) = U_x e. A {(F` x)})
Theorem	funimass4 3874	Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)
|- ((Fun F /\ A (_ dom F) -> ((F"A) (_ B <-> A.x e. A (F` x) e. B))
Theorem	fvelima 3875	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 3876	Function value in an image.
|- ((F Fn A /\ B (_ A) -> (C e. (F"B) <-> E.x e. B (F` x) = C))
Theorem	fniinfv 3877	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 3878	Singleton of function value.
|- ((F Fn A /\ B e. A) -> {(F` B)} = (F"{B}))
Theorem	ssimaex 3879	The existence of a subimage.
|- A e. V   =>   |- ((Fun F /\ B (_ (F"A)) -> E.x(x (_ A /\ B = (F"x)))
Theorem	ssimaexg 3880	The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
|- ((A e. C /\ Fun F /\ B (_ (F"A)) -> E.x(x (_ A /\ B = (F"x)))
Theorem	funfv 3881	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 3882	The value of a function. Definition of function value in [Enderton] p. 43.
|- (Fun F -> (F` A) = U.{y | AFy})
Theorem	funfv2f 3883	The value of a function. Version of funfv2 3882 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	dmfco 3884	Domains of a function composition.
|- ((Fun G /\ A e. dom G) -> (A e. dom ( F o. G) <-> (G` A) e. dom F))
Theorem	fvco 3885	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 3886	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 3887	Value of a function composition.
|- ((Fun F /\ G:A-->B /\ C e. A) -> ((F o. G)` C) = (F` (G` C)))
Theorem	fvopab3 3888	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 3889	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 3890	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 3891	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 3892	Value of a function given by an ordered-pair class abstraction. This version of fvopab4g 3890 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 3893	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 3894	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 3895	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 3896	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 3897	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 3896.
|- (x = A -> B = C)   =>   |- (-. C e. V -> ({<.x, y>. | y = B}` A) = (/))
Theorem	fvopabgf 3898	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 3899	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 3897 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 3900	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