mmtheorems38 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3701-3800 - Page 38 of 108

Type	Label	Description
Statement
Theorem	f1ocnv 3701	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 3702	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 3703	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 3704	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 3705	Pre-image of an image.
|- ((F:A-1-1->B /\ C (_ A) -> (`'F"(F"C)) = C)
Theorem	f1oun 3706	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	f1oco 3707	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 3708	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 3709	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 3710	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 2693.
|- ((F:A-1-1->B /\ B e. C) -> A e. V)
Theorem	ffoss 3711	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 3712	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 3713	The empty set maps one-to-one into any class.
|- (/):(/)-1-1->A
Theorem	f1o00 3714	One-to-one onto mapping of the empty set.
|- (F:(/)-1-1-onto->A <-> (F = (/) /\ A = (/)))
Theorem	fo00 3715	Onto mapping of the empty set.
|- (F:(/)-onto->A <-> (F = (/) /\ A = (/)))
Theorem	f1o0 3716	One-to-one onto mapping of the empty set.
|- (/):(/)-1-1-onto->(/)
Theorem	f1oi 3717	A restriction of the identity relation is a one-to-one onto function.
|- (I |` A):A-1-1-onto->A
Theorem	f1ovi 3718	The identity relation is a one-to-one onto function on the universe.
|- I:V-1-1-onto->V
Theorem	f1osn 3719	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 3720	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 3721	A function's value at a proper class is the empty set.
|- (-. A e. V -> (F` A) = (/))
Theorem	elfv 3722	Membership in a function value.
|- B e. V   =>   |- (A e. (F` B) <-> E.x(A e. x /\ A.y(BFy <-> y = x)))
Theorem	fveq1 3723	Equality theorem for function value.
|- (F = G -> (F` A) = (G` A))
Theorem	fveq2 3724	Equality theorem for function value.
|- (A = B -> (F` A) = (F` B))
Theorem	fveq1i 3725	Equality inference for function value.
|- F = G   =>   |- (F` A) = (G` A)
Theorem	fveq1d 3726	Equality deduction for function value.
|- (ph -> F = G)   =>   |- (ph -> (F` A) = (G` A))
Theorem	fveq2i 3727	Equality inference for function value.
|- A = B   =>   |- (F` A) = (F` B)
Theorem	fveq2d 3728	Equality deduction for function value.
|- (ph -> A = B)   =>   |- (ph -> (F` A) = (F` B))
Theorem	hbfv 3729	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 3730	Deduction version of bound-variable hypothesis builder hbfv 3729. If a closed theorem version is desired, see hbfvd2 3731.
|- (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 3731	Deduction version of bound-variable hypothesis builder hbfv 3729. This variant of hbfvd 3730 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 3732	The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27.
|- (F` A) e. V
Theorem	fv3 3733	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 3734	The value of a restricted function.
|- (A e. B -> ((F |` B)` A) = (F` A))
Theorem	funssfv 3735	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 3736	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27.
|- A e. V   =>   |- ((AFy /\ E!y AFy) -> (F` A) = y)
Theorem	tz6.12 3737	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 3738	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 3739	Function value when F is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27.
|- (-. E!y AFy -> (F` A) = (/))
Theorem	tz6.12c 3740	Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27.
|- A e. V   =>   |- (E!y AFy -> ((F` A) = y <-> AFy))
Theorem	tz6.12i 3741	Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27.
|- A e. V   =>   |- (B =/= (/) -> ((F` A) = B -> AFB))
Theorem	csbfv12g 3742	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 3743	Move class substitution in and out of a function value.
|- (A e. C -> [_A / x]_(F` B) = (F` [_A / x]_B))
Theorem	csbfvg 3744	Substitution for a function value.
|- (A e. C -> [_A / x]_(F` x) = (F` A))
Theorem	ndmfv 3745	The value of a class outside its domain is the empty set.
|- (-. A e. dom F -> (F` A) = (/))
Theorem	ndmfvrcl 3746	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 3747	If a function value has a member, the argument belongs to the domain.
|- (A e. (F` B) -> B e. dom F)
Theorem	nfvres 3748	A non-element of a restriction has empty value.
|- (-. A e. B -> ((F |` B)` A) = (/))
Theorem	fveqres 3749	Equal values imply equal values in a restriction.
|- ((F` A) = (G` A) -> ((F |` B)` A) = ((G |` B)` A))
Theorem	funbrfv 3750	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 3751	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 3752	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 3753	Equivalence of function value and binary relation.
|- C e. V   =>   |- ((F Fn A /\ B e. A) -> ((F` B) = C <-> BFC))
Theorem	fnopfvb 3754	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 3755	Equivalence of function value and binary relation.
|- B e. V   =>   |- ((Fun F /\ A e. dom F) -> ((F` A) = B <-> AFB))
Theorem	funopfvb 3756	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 3757	Function value in terms of a binary relation.
|- ((Fun F /\ A e. dom F /\ B e. C) -> ((F` A) = B <-> AFB))
Theorem	fnopabfv 3758	Representation of a function in terms of its values.
|- (F Fn A <-> F = {<.x, y>. | (x e. A /\ y = (F` x))})
Theorem	fnrnfv 3759	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)}