mmtheorems39 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3801-3900 - Page 39 of 107

Type	Label	Description
Statement
Theorem	funimass5 3801	A subclass of a preimage in terms of function values.
|- ((Fun F /\ A (_ dom F) -> (A (_ (`'F"B) <-> A.x e. A (F` x) e. B))
Theorem	funconstss 3802	Two ways of specifying that a function is constant on a subdomain.
|- ((Fun F /\ A (_ dom F) -> (A.x e. A (F` x) = B <-> A (_ (`'F"{B})))
Theorem	fvimacnvALT 3803	Another proof of fvimacnv 3799, based on funimass3 3800. If funimass3 3800 is ever proved directly, as opposed to using funimacnv 3566 pointwise, then the proof of funimacnv 3566 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 3804	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 3805	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 3806	A function's value belongs to its range.
|- ((Fun F /\ A e. dom F) -> (F` A) e. ran F)
Theorem	fnfvelrn 3807	A function's value belongs to its range.
|- ((F Fn A /\ B e. A) -> (F` B) e. ran F)
Theorem	ffvelrn 3808	A function's value belongs to its codomain.
|- ((F:A-->B /\ C e. A) -> (F` C) e. B)
Theorem	ffvelrni 3809	A function's value belongs to its codomain.
|- F:A-->B   =>   |- (C e. A -> (F` C) e. B)
Theorem	dff4 3810	Alternate definition of a mapping.
|- (F:A-->B <-> (F Fn A /\ F (_ (A X. B)))
Theorem	dff2 3811	Alternate definition of a mapping.
|- (F:A-->B <-> (F (_ (A X. B) /\ A.x e. A E!y xFy))
Theorem	dff3 3812	Alternate definition of a mapping.
|- (F:A-->B <-> (F (_ (A X. B) /\ A.x e. A E!y e. B xFy))
Theorem	dffo3 3813	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 3814	Alternate definition of an onto mapping.
|- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x e. A xFy))
Theorem	dffo5 3815	Alternate definition of an onto mapping.
|- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x xFy))
Theorem	exfo 3816	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 3817	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 3818	Inclusion of a function in a cross product.
|- F = {<.x, y>. | (x e. A /\ y = C)}   =>   |- (A.x e. A C e. B -> F (_ (A X. B))
Theorem	rnssopab 3819	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 (_ B)
Theorem	fopab3 3820	Functionality of an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = C)}   &   |- C e. V   =>   |- (ran F (_ B <-> F:A-->B)
Theorem	fopab 3821	Functionality of an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = C)}   &   |- (x e. A -> C e. B)   =>   |- F:A-->B
Theorem	ffnfv 3822	A function maps to a class to which all values belong.
|- (F:A-->B <-> (F Fn A /\ A.x e. A (F` x) e. B))
Theorem	ffnfvf 3823	A function maps to a class to which all values belong. This version of ffnfv 3822 uses bound-variable hypotheses instead of distinct variable conditions.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   &   |- (y e. F -> A.x y e. F)   =>   |- (F:A-->B <-> (F Fn A /\ A.x e. A (F` x) e. B))
Theorem	fnfvrnss 3824	An upper bound for range determined by function values.
|- ((F Fn A /\ A.x e. A (F` x) e. B) -> ran F (_ B)
Theorem	fopabfv 3825	Representation of a mapping in terms of its values.
|- (F:A-->B <-> (F = {<.x, y>. | (x e. A /\ y = (F` x))} /\ A.x e. A (F` x) e. B))
Theorem	fopabco 3826	Composition of two functions expressed as ordered-pair class abstractions. Note that v may be assigned to w, y, or z if desired.
|- R e. V   &   |- S e. V   &   |- T e. V   &   |- (z = R -> S = T)   &   |- F = {<.x, y>. | (x e. A /\ y = R)}   &   |- G = {<.z, w>. | (z e. B /\ w = S)}   &   |- H = {<.x, v>. | (x e. A /\ v = T)}   =>   |- (ran F (_ B -> (G o. F) = H)
Theorem	fopabcos 3827	Composition of two functions expressed as ordered-pair class abstractions.
|- C e. V   &   |- D e. V   &   |- F = {<.x, y>. | (x e. A /\ y = C)}   &   |- G = {<.x, y>. | (x e. B /\ y = D)}   =>   |- (ran G (_ A -> (F o. G) = {<.x, y>. | (x e. B /\ y = [_D / x]_C)})
Theorem	fsn 3828	A function maps a singleton to a singleton iff it is the singleton of a ordered pair.
|- A e. V   &   |- B e. V   =>   |- (F:{A}-->{B} <-> F = {<.A, B>.})
Theorem	xpsn 3829	The cross product of two singletons.
|- A e. V   &   |- B e. V   =>   |- ({A} X. {B}) = {<.A, B>.}
Theorem	fsn2 3830	A function that maps a singleton to a class is the singleton of an ordered pair.
|- A e. V   =>   |- (F:{A}-->B <-> ((F` A) e. B /\ F = {<.A, (F` A)>.}))
Theorem	fnressn 3831	A function restricted to a singleton.
|- ((F Fn A /\ B e. A) -> (F |` {B}) = {<.B, (F` B)>.})
Theorem	fressnfv 3832	The value of a function restricted to a singleton.
|- ((F Fn A /\ B e. A) -> ((F |` {B}):{B}-->C <-> (F` B) e. C))
Theorem	fvconst 3833	The value of a constant function.
|- ((F:A-->{B} /\ C e. A) -> (F` C) = B)
Theorem	fopabsn 3834	The singleton of an ordered pair expressed as an ordered pair class abstraction.
|- A e. V   &   |- B e. V   =>   |- {<.A, B>.} = {<.x, y>. | (x e. {A} /\ y = B)}
Theorem	fopabap 3835	Append an additional value to a function.
|- A e. V   &   |- B e. V   &   |- (R u. {A}) = S   &   |- (x = A -> C = B)   =>   |- ({<.x, y>. | (x e. R /\ y = C)} u. {<.A, B>.}) = {<.x, y>. | (x e. S /\ y = C)}
Theorem	fvi 3836	The value of the identity function.
|- (A e. B -> (I` A) = A)
Theorem	fvresi 3837	The value of a restricted identity function.
|- (B e. A -> ((I |` A)` B) = B)
Theorem	fvconst2g 3838	The value of a constant function.
|- ((B e. D /\ C e. A) -> ((A X. {B})` C) = B)
Theorem	fconst2g 3839	A constant function expressed as a cross product.
|- (B e. C -> (F:A-->{B} <-> F = (A X. {B})))
Theorem	fvconst2 3840	The value of a constant function.
|- B e. V   =>   |- (C e. A -> ((A X. {B})` C) = B)
Theorem	fconst2 3841	A constant function expressed as a cross product.
|- B e. V   =>   |- (F:A-->{B} <-> F = (A X. {B}))
Theorem	fconst5 3842	Two ways to express that a function is constant.
|- ((F Fn A /\ A =/= (/)) -> (F = (A X. {B}) <-> ran F = {B}))
Theorem	fconstfv 3843	A constant function expressed in terms of its functionality, domain, and value. See also fconst2 3841.
|- (F:A-->{B} <-> (F Fn A /\ A.x e. A (F` x) = B))
Theorem	fconst3 3844	Two ways to express a constant function.
|- (F:A-->{B} <-> (F Fn A /\ A (_ (`'F"{B})))
Theorem	fconst4 3845	Two ways to express a constant function.
|- (F:A-->{B} <-> (F Fn A /\ (`'F"{B}) = A))
Theorem	funfvima 3846	A function's value in a pre-image belongs to the image.
|- ((Fun F /\ B e. dom F) -> (B e. A -> (F` B) e. (F"A)))
Theorem	funfvima2 3847	A function's value in an included pre-image belongs to the image.
|- ((Fun F /\ A (_ dom F) -> (B e. A -> (F` B) e. (F"A)))
Theorem	funfvima3 3848	A class including a function contains the function's value in the image of the singleton of the argument.
|- ((Fun F /\ F (_ G) -> (A e. dom F -> (F