Metamath Proof Explorer - mmtheorems29

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Statement List for Metamath Proof Explorer - 2801-2900 - Page 29 of 58

Type	Label	Description
Statement
Theorem	f1ofn 2801	A one-to-one onto mapping is function on its domain.
|- (F:A-1-1-onto->B -> F Fn A)
Theorem	f1ofun 2802	A one-to-one onto mapping is a function.
|- (F:A-1-1-onto->B -> Fun F)
Theorem	f1orel 2803	A one-to-one onto mapping is a relation.
|- (F:A-1-1-onto->B -> Rel F)
Theorem	f1o2 2804	Alternate definition of one-to-one onto function.
|- (F:A-1-1-onto->B <-> (F Fn A /\ Fun `'F /\ ran F = B))
Theorem	f1o3 2805	Alternate definition of one-to-one onto function.
|- (F:A-1-1-onto->B <-> (F:A-onto->B /\ Fun `'F))
Theorem	f1ofo 2806	A one-to-one onto function is an onto function.
|- (F:A-1-1-onto->B -> F:A-onto->B)
Theorem	f1o4 2807	Alternate definition of one-to-one onto function.
|- (F:A-1-1-onto->B <-> (F Fn A /\ `'F Fn B))
Theorem	f1o5 2808	Alternate definition of one-to-one onto function.
|- (F:A-1-1-onto->B <-> (F:A-1-1->B /\ ran F = B))
Theorem	f1orn 2809	A one-to-one function maps onto its range.
|- (F:A-1-1-onto->ran F <-> (F Fn A /\ Fun `'F))
Theorem	f1f1orn 2810	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	f1ocnv 2811	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 2812	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 2813	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	f1imacnv 2814	Converse image of an image.
|- ((F:A-1-1->B /\ C (_ A) -> (`'F"(F"C)) = C)
Theorem	f1oun 2815	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 2816	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 2817	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 2818	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 2819	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 1075.
|- (B e. C -> (F:A-1-1->B -> A e. V))
Theorem	ffoss 2820	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 2821	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 2822	The empty set maps one-to-one into any class.
|- (/):(/)-1-1->A
Theorem	f1o00 2823	One-to-one onto mapping of the empty set.
|- (F:(/)-1-1-onto->A <-> (F = (/) /\ A = (/)))
Theorem	f1o0 2824	One-to-one onto mapping of the empty set.
|- (/):(/)-1-1-onto->(/)
Theorem	f1oi 2825	A restriction of the identity relation is a one-to-one onto function.
|- (I |` A):A-1-1-onto->A
Theorem	f1ovi 2826	The identity relation is a one-to-one onto function on the universe.
|- I:V-1-1-onto->V
Theorem	f1osn 2827	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 2828	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 2829	A function's value at a proper class is the empty set.
|- (-. A e. V -> (F` A) = (/))
Theorem	elfv 2830	Membership in a function value.
|- B e. V   =>   |- (A e. (F` B) <-> E.x(A e. x /\ A.y(BFy <-> y = x)))
Theorem	fveq1 2831	Equality theorem for function value.
|- (F = G -> (F` A) = (G` A))
Theorem	fveq2 2832	Equality theorem for function value.
|- (A = B -> (F` A) = (F` B))
Theorem	fveq1i 2833	Equality inference for function value.
|- F = G   =>   |- (F` A) = (G` A)
Theorem	fveq1d 2834	Equality deduction for function value.
|- (ph -> F = G)   =>   |- (ph -> (F` A) = (G` A))
Theorem	fveq2i 2835	Equality inference for function value.
|- A = B   =>   |- (F` A) = (F` B)
Theorem	fveq2d 2836	Equality deduction for function value.
|- (ph -> A = B)   =>   |- (ph -> (F` A) = (F` B))
Theorem	hbfv 2837	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	fvex 2838	The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27.
|- (F` A) e. V
Theorem	fv3 2839	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 2840	The value of a restricted function.
|- (A e. B -> ((F |` B)` A) = (F` A))
Theorem	funssfv 2841	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 2842	Theorem 6.12(1) of [TakeutiZaring] p. 27.
|- A e. V   =>   |- ((AFy /\ E!y AFy) -> (F` A) = y)
Theorem	tz6.12 2843	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 2844	Function value requiring only that y not be 'free' in F (but not necessarily absent from it).
|- (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 2845	Theorem 6.12(2) of [TakeutiZaring] p. 27.
|- (-. E!y AFy -> (F` A) = (/))
Theorem	tz6.12c 2846	Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27.
|- A e. V   =>   |- (E!y AFy -> ((F` A) = y <-> AFy))
Theorem	tz6.12i 2847	Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27.
|- A e. V   =>   |- (-. B = (/) -> ((F` A) = B -> AFB))
Theorem	ndmfv 2848	The value of a class outside its domain is the empty set.
|- (-. A e. dom F -> (F` A) = (/))
Theorem	ndmfvrcl 2849	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	nfvres 2850	A non-element of a restriction has empty value.
|- (-. A e. B -> ((F |` B)` A) = (/))
Theorem	fveqres 2851	Equal values imply equal values in a restriction.
|- ((F` A) = (G` A) -> ((F |` B)` A) = ((G |` B)` A))
Theorem	funbrfv 2852	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	funfvopi 2853	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 2854	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	fnfvbr 2855	Equivalence of function value and binary relation.
|- C e. V   =>   |- ((F Fn A /\ B e. A) -> ((F` B) = C <-> BFC))
Theorem	fnfvop 2856	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	funfvop 2857	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	fnopabfv 2858	Representation of a function in terms of its values.
|- (F Fn A <-> F = {<.x, y>. | (x e. A /\ y = (F` x))})
Theorem	fvelima 2859	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	fniunfv 2860	The indexed union of a function's values is the union of its range.
|- (F Fn A -> U.x e. A (F` x) = U.ran F)
Theorem	fnsnfv 2861	Singleton of function value.
|- ((F Fn A /\ B e. A) -> {(F` B)} = (F"{B}))
Theorem	funfv 2862	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 2863	The value of a function. Definition of function value in [Enderton] p. 43.
|- (Fun F -> (F` A) = U.{y | <.A, y>. e. F})
Theorem	dmfco 2864	Domains of a function composition.
|- ((Fun G /\ A e. dom G) -> (A e. dom (F o. G) <-> (G` A) e. dom F))
Theorem	fvco 2865	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 2866	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 2867	Value of a function composition.
|- (((Fun F /\ G:A-->B) /\ C e. A) -> ((F o. G)` C) = (F` (G` C)))
Theorem	fvopab3 2868	Value of a function given by ordered pair abstraction.
|- B e. V   &   |- (x = A ->