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	fvimacnvALT 3801	Another proof of fvimacnv 3797, based on funimass3 3798. If funimass3 3798 is ever proved directly, as opposed to using funimacnv 3564 pointwise, then the proof of funimacnv 3564 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 3802	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 3803	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 3804	A function's value belongs to its range.
|- ((Fun F /\ A e. dom F) -> (F` A) e. ran F)
Theorem	fnfvelrn 3805	A function's value belongs to its range.
|- ((F Fn A /\ B e. A) -> (F` B) e. ran F)
Theorem	ffvelrn 3806	A function's value belongs to its codomain.
|- ((F:A-->B /\ C e. A) -> (F` C) e. B)
Theorem	ffvelrni 3807	A function's value belongs to its codomain.
|- F:A-->B   =>   |- (C e. A -> (F` C) e. B)
Theorem	dff4 3808	Alternate definition of a mapping.
|- (F:A-->B <-> (F Fn A /\ F (_ (A X. B)))
Theorem	dff2 3809	Alternate definition of a mapping.
|- (F:A-->B <-> (F (_ (A X. B) /\ A.x e. A E!y xFy))
Theorem	dff3 3810	Alternate definition of a mapping.
|- (F:A-->B <-> (F (_ (A X. B) /\ A.x e. A E!y e. B xFy))
Theorem	dffo3 3811	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 3812	Alternate definition of an onto mapping.
|- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x e. A xFy))
Theorem	dffo5 3813	Alternate definition of an onto mapping.
|- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x xFy))
Theorem	exfo 3814	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 3815	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 3816	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 3817	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 3818	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 3819	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 3820	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 3821	A function maps to a class to which all values belong. This version of ffnfv 3820 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 3822	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 3823	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 3824	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 3825	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 3826	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 3827	The cross product of two singletons.
|- A e. V   &   |- B e. V   =>   |- ({A} X. {B}) = {<.A, B>.}
Theorem	fsn2 3828	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 3829	A function restricted to a singleton.
|- ((F Fn A /\ B e. A) -> (F |` {B}) = {<.B, (F` B)>.})
Theorem	fressnfv 3830	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 3831	The value of a constant function.
|- ((F:A-->{B} /\ C e. A) -> (F` C) = B)
Theorem	fopabsn 3832	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 3833	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 3834	The value of the identity function.
|- (A e. B -> (I` A) = A)
Theorem	fvresi 3835	The value of a restricted identity function.
|- (B e. A -> ((I |` A)` B) = B)
Theorem	fvconst2g 3836	The value of a constant function.
|- ((B e. D /\ C e. A) -> ((A X. {B})` C) = B)
Theorem	fconst2g 3837	A constant function expressed as a cross product.
|- (B e. C -> (F:A-->{B} <-> F = (A X. {B})))
Theorem	fvconst2 3838	The value of a constant function.
|- B e. V   =>   |- (C e. A -> ((A X. {B})` C) = B)
Theorem	fconst2 3839	A constant function expressed as a cross product.
|- B e. V   =>   |- (F:A-->{B} <-> F = (A X. {B}))
Theorem	fconst5 3840	Two ways to express that a function is constant.
|- ((F Fn A /\ A =/= (/)) -> (F = (A X. {B}) <-> ran F = {B}))
Theorem	fconstfv 3841	A constant function expressed in terms of its functionality, domain, and value. See also fconst2 3839.
|- (F:A-->{B} <-> (F Fn A /\ A.x e. A (F` x) = B))
Theorem	fconst3 3842	Two ways to express a constant function.
|- (F:A-->{B} <-> (F Fn A /\ A (_ (`'F"{B})))
Theorem	fconst4 3843	Two ways to express a constant function.
|- (F:A-->{B} <-> (F Fn A /\ (`'F"{B}) = A))
Theorem	funfvima 3844	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 3845	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 3846	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` A) e. (G"{A})))
Theorem	fvclss 3847	Upper bound for the class of values of a class.
|- {y | E.x y = (F` x)} (_ (ran F u. {(/)})
Theorem	fvclex 3848	Existence of the class of values of a set.
|- F e. V   =>   |- {y | E.x y = (F` x)} e. V
Theorem	fvresex 3849	Existence of the class of values of a restricted class.
|- A e. V   =>   |- {y |