mmtheorems40 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3901-4000 - Page 40 of 123

Type	Label	Description
Statement
Theorem	fvopab 3901	The value of a function given by an ordered-pair class abstraction.
|- A e. V   &   |- C e. V   &   |- (x = A -> B = C)   =>   |- ({<.x, y>. | y = B}` A) = C
Theorem	fvopab2 3902	Value of a function given by an ordered-pair class abstraction.
|- ((x e. A /\ B e. C) -> ({<.x, y>. | (x e. A /\ y = B)}` x) = B)
Theorem	fvopabs 3903	The value of a function given by an ordered-pair class abstraction, using class substitution.
|- A e. V   &   |- B e. V   =>   |- ({<.x, y>. | y = B}` A) = [_A / x]_B
Theorem	fvopab5 3904	The value of a function that is expressed as an ordered pair abstraction.
|- F = {<.x, y>. | ph}   &   |- (x = A -> (ph <-> ps))   =>   |- ((Fun F /\ A e. B) -> (F` A) = U.{y | ps})
Theorem	fvopab6 3905	Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- F = {<.x, y>. | (ph /\ y = B)}   &   |- (x = A -> (ph <-> ps))   &   |- (x = A -> B = C)   =>   |- ((A e. D /\ C e. R /\ ps) -> (F` A) = C)
Theorem	fvsn 3906	The value of a singleton of an ordered pair is the second member.
|- A e. V   &   |- B e. V   =>   |- ({<.A, B>.}` A) = B
Theorem	fvsnun1 3907	The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 3908.
|- A e. V   &   |- B e. V   &   |- G = ({<.A, B>.} u. (F |` (C \ {A})))   =>   |- (G` A) = B
Theorem	fvsnun2 3908	The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 3907.
|- A e. V   &   |- B e. V   &   |- G = ({<.A, B>.} u. (F |` (C \ {A})))   =>   |- (D e. (C \ {A}) -> (G` D) = (F` D))
Theorem	eqfnfv 3909	Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28.
|- ((F Fn A /\ G Fn B) -> (F = G <-> (A = B /\ A.x e. A (F` x) = (G` x))))
Theorem	eqfnfvALT 3910	Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted).
|- ((F Fn A /\ G Fn A) -> (F = G <-> A.x e. A (F` x) = (G` x)))
Theorem	eqfnfv2 3911	Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted).
|- ((F Fn A /\ G Fn A) -> (F = G <-> A.x e. A (F` x) = (G` x)))
Theorem	eqfnfvf2 3912	Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv2 3911 uses bound-variable hypotheses instead of distinct variable conditions.
|- (y e. F -> A.x y e. F)   &   |- (y e. G -> A.x y e. G)   =>   |- ((F Fn A /\ G Fn A) -> (F = G <-> A.x e. A (F` x) = (G` x)))
Theorem	fvreseq 3913	Equality of restricted functions is determined by their values.
|- (((F Fn A /\ G Fn A) /\ B (_ A) -> ((F |` B) = (G |` B) <-> A.x e. B (F` x) = (G` x)))
Theorem	elrnopabg 3914	Membership in the range of an ordered pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = B)}   =>   |- (A.x e. A B e. D -> (C e. ran F <-> E.x e. A C = B))
Theorem	elrnopab 3915	Membership in the range of an ordered pair class abstraction.
|- B e. V   &   |- F = {<.x, y>. | (x e. A /\ y = B)}   =>   |- (C e. ran F <-> E.x e. A C = B)
Theorem	chfnrn 3916	The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain.
|- ((F Fn A /\ A.x e. A (F` x) e. x) -> ran F (_ U.A)
Theorem	funfvop 3917	Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41.
|- ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. F)
Theorem	fvimacnvi 3918	A member of a preimage is a function value argument.
|- ((Fun F /\ A e. (`'F"B)) -> (F` A) e. B)
Theorem	fvimacnv 3919	The argument of a function value belongs to the pre-image of any class containing the function value. (Contributed by Raph Levien, 20-Nov-2006. He remarks: "This proof is unsatisfying, because it seems to me that funimass2 3678 could probably be strengthened to a biconditional.")
|- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> A e. (`'F"B)))
Theorem	funimass3 3920	A kind of contraposition law that infers an image subclass from a subclass of a pre-image. (Contributed by Raph Levien, 20-Nov-2006. He remarks: "Likely this could be proved directly, and fvimacnv 3919 would be the special case of A being a singleton, but it works this way round too.")
|- ((Fun F /\ A (_ dom F) -> ((F"A) (_ B <-> A (_ (`'F"B)))
Theorem	funimass5 3921	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 3922	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 3923	Another proof of fvimacnv 3919, based on funimass3 3920. If funimass3 3920 is ever proved directly, as opposed to using funimacnv 3676 pointwise, then the proof of funimacnv 3676 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 3924	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 3925	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 3926	A function's value belongs to its range.
|- ((Fun F /\ A e. dom F) -> (F` A) e. ran F)
Theorem	fnfvelrn 3927	A function's value belongs to its range.
|- ((F Fn A /\ B e. A) -> (F` B) e. ran F)
Theorem	ffvelrn 3928	A function's value belongs to its codomain.
|- ((F:A-->B /\ C e. A) -> (F` C) e. B)
Theorem	ffvelrni 3929	A function's value belongs to its codomain.
|- F:A-->B   =>   |- (C e. A -> (F` C) e. B)
Theorem	dff2 3930	Alternate definition of a mapping.
|- (F:A-->B <-> (F Fn A /\ F (_ (A X. B)))
Theorem	dff3 3931	Alternate definition of a mapping.
|- (F:A-->B <-> (F (_ (A X. B) /\ A.x e. A E!y xFy))
Theorem	dff4 3932	Alternate definition of a mapping.
|- (F:A-->B <-> (F (_ (A X. B) /\ A.x e. A E!y e. B xFy))
Theorem	dffo3 3933	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 3934	Alternate definition of an onto mapping.
|- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x e. A xFy))
Theorem	dffo5 3935	Alternate definition of an onto mapping.
|- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x xFy))
Theorem	exfo 3936	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 3937	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 3938	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 3939	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 3940	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 3941	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 3942	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 3943	A function maps to a class to which all values belong. This version of ffnfv 3942 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 3944	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 3945	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 3946	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 3947	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 3948	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 3949	The cross product of two singletons.
|- A e. V   &   |- B e. V   =>   |- ({A} X. {B}) = {<.A, B>.}
Theorem	fsn2 3950	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 3951	A function restricted to a singleton.
|- ((F Fn A /\ B e. A) -> (F |` {B}) = {<.B, (F` B)>.})
Theorem	fressnfv 3952	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 3953	The value of a constant function.
|- ((F:A-->{B} /\ C e. A) -> (F` C) = B)
Theorem	fopabsn 3954	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 3955	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 3956	The value of the identity function.
|- (A e. B -> (I` A) = A)
Theorem	fvresi 3957	The value of a restricted identity function.
|- (B e. A -> ((I |` A)` B) = B)
Theorem	fvconst2g 3958	The value of a constant function.
|- ((B e. D /\ C e. A) -> ((A X. {B})` C) = B)
Theorem	fconst2g 3959	A constant function expressed as a cross product.
|- (B e. C -> (F:A-->{B} <-> F = (A X. {B})))
Theorem	fvconst2 3960	The value of a constant function.
|- B e. V   =>   |- (C e. A -> ((A X. {B})` C) = B)
Theorem	fconst2 3961	A constant function expressed as a cross product.
|- B e. V   =>   |- (F:A-->{B} <-> F = (A X. {B}))
Theorem	fconst5 3962	Two ways to express that a function is constant.
|- ((F Fn A /\ A =/= (/)) -> (F = (A X. {B}) <-> ran F = {B}))
Theorem	fconstfv 3963	A constant function expressed in terms of its functionality, domain, and value. See also fconst2 3961.
|- (F:A-->{B} <-> (F Fn A /\ A.x e. A (F` x) = B))
Theorem	fconst3 3964	Two ways to express a constant function.
|- (F:A-->{B} <-> (F Fn A /\ A (_ (`'F"{B})))
Theorem	fconst4 3965	Two ways to express a constant function.
|- (F:A-->{B} <-> (F Fn A /\ (`'F"{B}) = A))
Theorem	funfvima 3966	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 3967	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 3968	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 3969	Upper bound for the class of values of a class.
|- {y | E.x y = (F` x)} (_ (ran F u. {(/)})
Theorem	fvclex 3970	Existence of the class of values of a set.
|- F e. V   =>   |- {y | E.x y = (F` x)} e. V
Theorem	fvresex 3971	Existence of the class of values of a restricted class.
|- A e. V   =>   |- {y | E.x y = ((F |` A)` x)} e. V
Theorem	abrexexlem1 3972	Lemma for abrexex 3974. Shows the existence of a class of existentially restricted function values.
Theorem	abrexexlem2 3973	Lemma for abrexex 3974. Almost there, but still requires that B be a set.
Theorem	abrexex 3974	Existence of a class abstraction of existentially restricted sets. x is normally a free-variable parameter in the class expression substituted for B, which can be thought of as B(x). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path abrexexlem2 3973, abrexexlem1 3972, fvresex 3971, resfunexg 3685, and funimaexg 3681. See also abrexex2 3985.
|- A e. V   =>   |- {y | E.x e. A y = B} e. V
Theorem	abrexexg 3975	Existence of a class abstraction of existentially restricted sets. x is normally a free-variable parameter in B. The antecedent assures us that A is a set.
|- (A e. C -> {y | E.x e. A y = B} e. V)
Theorem	iunexg 3976	The existence of an indexed union. x is normally a free-variable parameter in B.
|- ((A e. C /\ A.x e. A B e. D) -> U_x e. A B e. V)
Theorem	iunex 3977	The existence of an indexed union. x is normally a free-variable parameter in the class expression substituted for B, which can be read informally as B(x).
|- A e. V   &   |- B e. V   =>   |- U_x e. A B e. V
Theorem	imaiun 3978	The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50.
|- (A"U.B) = U_x e. B (A"x)
Theorem	fniunfv 3979	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	funiunfv 3980	The indexed union of a function's values is the union of its image under the index class. Note: This theorem depends on the fact that our function value is the empty set outside of its domain. If the antecedent is changed to F Fn A, the theorem can be proved without this dependency.
|- (Fun F -> U_x e. A (F` x) = U.(F"A))
Theorem	eluniima 3981	Membership in the union of an image of a function.
|- (Fun F -> (B e. U.(F"A) <-> E.x e. A B e. (F` x)))
Theorem	elunirn 3982	Membership in the union of the range of a function.
|- (Fun F -> (A e. U.ran F <-> E.x e. dom F A e. (F` x)))
Theorem	elunirnALT 3983	Membership in the union of the range of a function, proved directly. Unlike elunirn 3982, it doesn't appeal to ndmfv 3856 (via funiunfv 3980).
|- (Fun F -> (A e. U.ran F <-> E.x e. dom F A e. (F` x)))
Theorem	funiunfvf 3984	The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 3980 uses a bound-variable hypothesis in place of a distinct variable condition.
|- (y e. F -> A.x y e. F)   =>   |- (Fun F -> U_x e. A (F` x) = U.(F"A))
Theorem	abrexex2 3985	Existence of an existentially restricted class abstraction. ph is normally has free-variable parameters x and y. This is a powerful generalization of abrexex 3974.
|- A e. V   &   |- {y | ph} e. V   =>   |- {y | E.x e. A ph} e. V
Theorem	abexssex 3986	Existence of a class abstraction with an existentially quantified expression. Both x and y can be free in ph.
|- A e. V   &   |- {y | ph} e. V   =>   |- {y | E.x(x (_ A /\ ph)} e. V
Theorem	abexex 3987	A condition where a class builder continues to exist after its wff is existentially quantified.
|- A e. V   &   |- (ph -> x e. A)   &   |- {y | ph} e. V   =>   |- {y | E.xph} e. V
Theorem	dff13 3988	A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43.
|- (F:A-1-1->B <-> (F:A-->B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))
Theorem	dff13f 3989	A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43.
|- (z e. F -> A.x z e. F)   &   |- (z e. F -> A.y z e. F)   =>   |- (F:A-1-1->B <-> (F:A-->B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))
Theorem	f1fveq 3990	Equality of function values for a one-to-one function.
|- ((F:A-1-1->B /\ (C e. A /\ D e. A)) -> ((F` C) = (F` D) <-> C = D))
Theorem	dff1o6 3991	A one-to-one onto function in terms of function values.
|- (F:A-1-1-onto->B <-> (F Fn A /\ ran F = B /\ A.x e. A A.y e. A ((F` x) = (F` y) -> x = y)))
Theorem	f1ocnvfv1 3992	The converse value of the value of a one-to-one onto function.
|- ((F:A-1-1-onto->B /\ C e. A) -> (`'F` (F` C)) = C)
Theorem	f1ocnvfv2 3993	The value of the converse value of a one-to-one onto function.
|- ((F:A-1-1-onto->B /\ C e. B) -> (F` (`'F` C)) = C)
Theorem	f1ocnvfv 3994	Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.)
|- ((F:A-1-1-onto->B /\ C e. A) -> ((F` C) = D -> (`'F` D) = C))
Theorem	f1ocnvfvb 3995	Relationship between the value of a one-to-one onto function and the value of its converse.
|- ((F:A-1-1-onto->B /\ C e. A /\ D e. B) -> ((F` C) = D <-> (`'F` D) = C))
Theorem	f1ofveu 3996	There is one domain element for each value of a one-to-one onto function.
|- ((F:A-1-1-onto->B /\ C e. B) -> E!x e. A (F` x) = C)
Theorem	f1ocnvfv3 3997	Value of the converse of a one-to-one onto function.
|- ((F:A-1-1-onto->B /\ C e. B) -> (`'F` C) = U.{x e. A | (F` x) = C})
Theorem	f1ocnvdm 3998	The value of the converse of a one-to-one onto function belongs to its domain.
|- ((F:A-1-1-onto->B /\ C e. B) -> (`'F` C) e. A)
Theorem	cbvfo 3999	Change bound variable between domain and range of function.
|- ((F` x) = y -> (ph <-> ps))   =>   |- (F:A-onto->B -> (A.x e. A ph <-> A.y e. B ps))
Theorem	cbvexfo 4000	Change bound variable between domain and range of function.
|- ((F` x) = y -> (ph <-> ps))   =>   |- (F:A-onto->B -> (E.x e. A ph <-> E.y e. B ps))