mmtheorems36 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3501-3600 - Page 36 of 107

Type	Label	Description
Statement
Theorem	rnco 3501	The range of the composition of two classes.
|- ran ( A o. B) = ran ( A |` ran B)
Theorem	rnco2 3502	The range of the composition of two classes.
|- ran ( A o. B) = (A"ran B)
Theorem	dmco2 3503	The domain of a composition. Exercise 27 of [Enderton] p. 53.
|- dom ( A o. B) = (`'B"dom A)
Theorem	cocnvcnv1 3504	A composition is not affected by a double converse of its first argument.
|- (`'`'A o. B) = (A o. B)
Theorem	cocnvcnv2 3505	A composition is not affected by a double converse of its second argument.
|- (A o. `'`'B) = (A o. B)
Theorem	cores2 3506	Absorption of a reverse (preimage) restriction of the second member of a class composition.
|- (dom A (_ C -> (A o. `'(`'B |` C)) = (A o. B))
Theorem	co02 3507	Composition with the empty set. Theorem 20 of [Suppes] p. 63.
|- (A o. (/)) = (/)
Theorem	co01 3508	Composition with the empty set.
|- ((/) o. A) = (/)
Theorem	coi1 3509	Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36.
|- (Rel A -> (A o. I) = A)
Theorem	coi2 3510	Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36.
|- (Rel A -> (I o. A) = A)
Theorem	coass 3511	Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations.
|- ((A o. B) o. C) = (A o. (B o. C))
Theorem	relssdr 3512	A relation is included in the cross product of its domain and range. Exercise 4.12(t) of [Mendelson] p. 235.
|- (Rel A -> A (_ (dom A X. ran A))
Theorem	unielrel 3513	The membership relation for a relation is inherited by class union.
|- ((Rel R /\ A e. R) -> U.A e. U.R)
Theorem	relfld 3514	The double union of a relation is its field.
|- (Rel R -> U.U.R = (dom R u. ran R))
Theorem	unidmrnOLD 3515	The double union of the universal restriction of a class.
|- U.U.(A |` V) = (dom A u. ran A)
Theorem	unidmrn 3516	The double union of the converse of a class is its field.
|- U.U.`'A = (dom A u. ran A)
Theorem	unixp 3517	The double class union of a non-empty cross product is the union of it members.
|- ((A X. B) =/= (/) -> U.U.(A X. B) = (A u. B))
Theorem	unixp0 3518	A cross product is empty iff its union is empty.
|- ((A X. B) = (/) <-> U.(A X. B) = (/))
Theorem	cnvexg 3519	The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26.
|- (A e. B -> `'A e. V)
Theorem	cnvex 3520	The converse of a set is a set. Corollary 6.8(1) of [TakeutiZaring] p. 26.
|- A e. V   =>   |- `'A e. V
Theorem	relcnvexb 3521	A relation is a set iff its converse is a set. (Contributed by FL, 3-Mar-2007.)
|- (Rel R -> (R e. V <-> `'R e. V))
Theorem	cnvpo 3522	The converse of a partial order relation is a partial order relation.
|- (R Po A <-> `'R Po A)
Theorem	cnvso 3523	The converse of a strict order relation is a strict order relation.
|- (R Or A <-> `'R Or A)
Theorem	coexg 3524	The composition of two sets is a set.
|- ((A e. C /\ B e. D) -> (A o. B) e. V)
Theorem	coex 3525	The composition of two sets is a set.
|- A e. V   &   |- B e. V   =>   |- (A o. B) e. V
Theorem	dffun2 3526	Alternate definition of a function.
|- (Fun A <-> (Rel A /\ A.xA.yA.z((xAy /\ xAz) -> y = z)))
Theorem	dffun3 3527	Alternate definition of function.
|- (Fun A <-> (Rel A /\ A.xE.zA.y(xAy -> y = z)))
Theorem	dffun4 3528	Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24.
|- (Fun A <-> (Rel A /\ A.xA.yA.z((<.x, y>. e. A /\ <.x, z>. e. A) -> y = z)))
Theorem	dffun5 3529	Alternate definition of function.
|- (Fun A <-> (Rel A /\ A.xE.zA.y(<.x, y>. e. A -> y = z)))
Theorem	dffunmof 3530	Definition of function, using bound-variable hypotheses instead of distinct variable conditions.
|- (z e. A -> A.x z e. A)   &   |- (z e. A -> A.y z e. A)   =>   |- (Fun A <-> (Rel A /\ A.xE*y xAy))
Theorem	dffunmo 3531	Alternate definition of a function using "at most one" notation.
|- (Fun A <-> (Rel A /\ A.xE*y xAy))
Theorem	funmo 3532	A function has at most one value for each argument.
|- (Fun A -> E*y xAy)
Theorem	funrel 3533	A function is a relation.
|- (Fun A -> Rel A)
Theorem	funss 3534	Subclass theorem for function predicate.
|- (A (_ B -> (Fun B -> Fun A))
Theorem	funeq 3535	Equality theorem for function predicate.
|- (A = B -> (Fun A <-> Fun B))
Theorem	hbfun 3536	Bound-variable hypothesis builder for a function.
|- (y e. F -> A.x y e. F)   =>   |- (Fun F -> A.xFun F)
Theorem	funeu 3537	There is exactly one value of a function.
|- ((Fun F /\ xFy) -> E!y xFy)
Theorem	funeu2 3538	There is exactly one value of a function.
|- ((Fun F /\ <.x, y>. e. F) -> E!y<.x, y>. e. F)
Theorem	dffun6 3539	Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one." However, dffun7 3540 shows that it doesn't matter which meaning we pick.)
|- (Fun A <-> (Rel A /\ A.x e. dom AE*y xAy))
Theorem	dffun7 3540	Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun6 3539.
|- (Fun A <-> (Rel A /\ A.x e. dom AE!y xAy))
Theorem	dffun8 3541	Alternate definition of a function.
|- (Fun A <-> (Rel A /\ A.x e. dom AE*y(y e. ran A /\ xAy)))
Theorem	funfn 3542	An equivalence for the function predicate.
|- (Fun A <-> A Fn dom A)
Theorem	funsn 3543	A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65.
|- A e. V   &   |- B e. V   =>   |- Fun {<.A, B>.}
Theorem	fun0 3544	The empty set is a function. Theorem 10.3 of [Quine] p. 65.
|- Fun (/)
Theorem	funi 3545	The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65.
|- Fun I
Theorem	nfunv 3546	The universe is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
|- -. Fun V
Theorem	funop 3547	A Kuratowski ordered pair is a function only if its components are equal.
|- A e. V   &   |- B e. V   =>   |- (Fun <.A, B>. -> A = B)
Theorem	funopg 3548	A Kuratowski ordered pair is a function only if its components are equal.
|- ((B e. C /\ Fun <.A, B>.) -> A = B)
Theorem	funopab 3549	A class of ordered pairs is a function when there is at most one second member for each pair.
|- (Fun {<.x, y>. | ph} <-> A.xE*yph)
Theorem	funopabeq 3550	A class of ordered pairs of values is a function.
|- Fun {<.x, y>. | y = A}
Theorem	funco 3551	The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25.
|- ((Fun F /\ Fun G) -> Fun (F o. G))
Theorem	funres 3552	A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25.
|- (Fun F -> Fun (F |` A))
Theorem	funssres 3553	The restriction of a function to the domain of a subclass equals the subclass.
|- ((Fun F /\ G (_ F) -> (F |` dom G) = G)
Theorem	fun2ssres 3554	Equality of restrictions of a function and a subclass.
|- ((Fun F /\ G (_ F /\ A (_ dom G) -> (F |` A) = (G |` A))
Theorem	funun 3555	The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43.
|- (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) -> Fun (F u. G))
Theorem	funcnvcnv 3556	The double converse of a function is a function.
|- (Fun A -> Fun `'`'A)
Theorem	funcnv2 3557	A simpler equivalence for single-rooted (see funcnv 3558).
|- (Fun `'A <-> A.yE*x xAy)
Theorem	funcnv 3558	The converse of a class is a function iff the class is single-rooted, which means that for any y in the range of A there is at most one x such that xAy. Definition of single-rooted in [Enderton] p. 43. See funcnv2 3557 for a simpler version.
|- (Fun `'A <-> A.y e. ran AE*x xAy)
Theorem	funcnv3 3559	A condition showing a class is single-rooted. (See funcnv 3558).
|- (Fun `'A <-> A.y e. ran AE!x e. dom A xAy)
Theorem	fun2cnv 3560	The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that A is not necessarily a function.
|- (Fun `'`'A <-> A.xE*y xAy)
Theorem	svrelfun 3561	A single-valued relation is a function. (See fun2cnv 3560 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24.
|- (Fun A <-> (Rel A /\ Fun `'`'A))
Theorem	fncnv 3562	Single-rootedness (see funcnv 3558) of a class cut down by a cross product.
|- (`'(R i^i (A X. B)) Fn B <-> A.y e. B E!x e. A xRy)
Theorem	fun11 3563	Two ways of stating that A is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one (but not necessarily a function).
|- ((Fun `'`'A /\ Fun `'A) <-> A.xA.yA.zA.w((xAy /\ zAw) -> (x = z <-> y = w)))
Theorem	fununi 3564	The union of a chain (with respect to inclusion) of functions is a function.
|- (A.f e. A (Fun f /\ A.g e. A (f (_ g \/ g (_ f)) -> Fun U.A)
Theorem	funcnvuni 3565	The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 3558 for "single-rooted" definition.)
|- (A.f e. A (Fun `'f /\ A.g e. A (f (_ g \/ g (_ f)) -> Fun `'U.A)
Theorem	fun11uni 3566	The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function.
|- (A.f e. A ((Fun f /\ Fun `'f) /\ A.g e. A (f (_ g \/ g (_ f)) -> (Fun U.A /\ Fun `'U.A))
Theorem	funin 3567	The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53.
|- (Fun F -> Fun (F i^i G))
Theorem	funres11 3568	The restriction of a one-to-one function is one-to-one.
|- (Fun `'F -> Fun `'(F |` A))
Theorem	funcnvres 3569	The converse of a restricted function.
|- (Fun `'F -> `'(F |` A) = (`'F |` (F"A)))
Theorem	cnvresid 3570	Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
|- `'(I |` A) = (I |` A)
Theorem	funcnvres2 3571	The converse of a restriction of the converse of a function equals the function restricted to the image of its converse.
|- (Fun F -> `'(`'F |` A) = (F |` (`'F"A)))
Theorem	funimacnv 3572	The image of the pre-image of a function.
|- (Fun F -> (F"(`'F"A)) = (A i^i ran F))
Theorem	funimass1 3573	A kind of contraposition law that infers a subclass of an image from a pre-image subclass.
|- ((Fun F /\ A (_ ran F) -> ((`'F"A) (_ B -> A (_ (F"B)))
Theorem	funimass2 3574	A kind of contraposition law that infers an image subclass from a subclass of a pre-image.
|- ((Fun F /\ A (_ (`'F"B)) -> (F"A) (_ B)
Theorem	imadif 3575	The image of a difference is the difference of images.
|- (Fun `'F -> (F"(A \ B)) = ((F"A) \ (F"B