Theorem elxp5 5022

Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5021 when the double intersection does not create class existence problems (caused by int0 3780). (Contributed by NM, 1-Aug-2004.)

Assertion

Ref	Expression
elxp5	|- ( A e. ( B X. C ) <-> ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) )

Proof of Theorem elxp5

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2692	. 2 |- ( A e. ( B X. C ) -> A e. _V )
2		elex 2692	. . . 4 |- ( |^| |^| A e. B -> |^| |^| A e. _V )
3		elex 2692	. . . 4 |- ( U. ran { A } e. C -> U. ran { A } e. _V )
4	2, 3	anim12i 336	. . 3 |- ( ( |^| |^| A e. B /\ U. ran { A } e. C ) -> ( |^| |^| A e. _V /\ U. ran { A } e. _V ) )
5		opexg 4145	. . . . 5 |- ( ( |^| |^| A e. _V /\ U. ran { A } e. _V ) -> <. |^| |^| A , U. ran { A } >. e. _V )
6	5	adantl 275	. . . 4 |- ( ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. _V /\ U. ran { A } e. _V ) ) -> <. |^| |^| A , U. ran { A } >. e. _V )
7		eleq1 2200	. . . . 5 |- ( A = <. |^| |^| A , U. ran { A } >. -> ( A e. _V <-> <. |^| |^| A , U. ran { A } >. e. _V ) )
8	7	adantr 274	. . . 4 |- ( ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. _V /\ U. ran { A } e. _V ) ) -> ( A e. _V <-> <. |^| |^| A , U. ran { A } >. e. _V ) )
9	6, 8	mpbird 166	. . 3 |- ( ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. _V /\ U. ran { A } e. _V ) ) -> A e. _V )
10	4, 9	sylan2 284	. 2 |- ( ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) -> A e. _V )
11		elxp 4551	. . . 4 |- ( A e. ( B X. C ) <-> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) )
12		sneq 3533	. . . . . . . . . . . . . 14 |- ( A = <. x , y >. -> { A } = { <. x , y >. } )
13	12	rneqd 4763	. . . . . . . . . . . . 13 |- ( A = <. x , y >. -> ran { A } = ran { <. x , y >. } )
14	13	unieqd 3742	. . . . . . . . . . . 12 |- ( A = <. x , y >. -> U. ran { A } = U. ran { <. x , y >. } )
15		vex 2684	. . . . . . . . . . . . 13 |- x e. _V
16		vex 2684	. . . . . . . . . . . . 13 |- y e. _V
17	15, 16	op2nda 5018	. . . . . . . . . . . 12 |- U. ran { <. x , y >. } = y
18	14, 17	syl6req 2187	. . . . . . . . . . 11 |- ( A = <. x , y >. -> y = U. ran { A } )
19	18	pm4.71ri 389	. . . . . . . . . 10 |- ( A = <. x , y >. <-> ( y = U. ran { A } /\ A = <. x , y >. ) )
20	19	anbi1i 453	. . . . . . . . 9 |- ( ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( ( y = U. ran { A } /\ A = <. x , y >. ) /\ ( x e. B /\ y e. C ) ) )
21		anass 398	. . . . . . . . 9 |- ( ( ( y = U. ran { A } /\ A = <. x , y >. ) /\ ( x e. B /\ y e. C ) ) <-> ( y = U. ran { A } /\ ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) )
22	20, 21	bitri 183	. . . . . . . 8 |- ( ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( y = U. ran { A } /\ ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) )
23	22	exbii 1584	. . . . . . 7 |- ( E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> E. y ( y = U. ran { A } /\ ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) )
24		snexg 4103	. . . . . . . . . 10 |- ( A e. _V -> { A } e. _V )
25		rnexg 4799	. . . . . . . . . 10 |- ( { A } e. _V -> ran { A } e. _V )
26	24, 25	syl 14	. . . . . . . . 9 |- ( A e. _V -> ran { A } e. _V )
27		uniexg 4356	. . . . . . . . 9 |- ( ran { A } e. _V -> U. ran { A } e. _V )
28	26, 27	syl 14	. . . . . . . 8 |- ( A e. _V -> U. ran { A } e. _V )
29		opeq2 3701	. . . . . . . . . . 11 |- ( y = U. ran { A } -> <. x , y >. = <. x , U. ran { A } >. )
30	29	eqeq2d 2149	. . . . . . . . . 10 |- ( y = U. ran { A } -> ( A = <. x , y >. <-> A = <. x , U. ran { A } >. ) )
31		eleq1 2200	. . . . . . . . . . 11 |- ( y = U. ran { A } -> ( y e. C <-> U. ran { A } e. C ) )
32	31	anbi2d 459	. . . . . . . . . 10 |- ( y = U. ran { A } -> ( ( x e. B /\ y e. C ) <-> ( x e. B /\ U. ran { A } e. C ) ) )
33	30, 32	anbi12d 464	. . . . . . . . 9 |- ( y = U. ran { A } -> ( ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) )
34	33	ceqsexgv 2809	. . . . . . . 8 |- ( U. ran { A } e. _V -> ( E. y ( y = U. ran { A } /\ ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) <-> ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) )
35	28, 34	syl 14	. . . . . . 7 |- ( A e. _V -> ( E. y ( y = U. ran { A } /\ ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) <-> ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) )
36	23, 35	syl5bb 191	. . . . . 6 |- ( A e. _V -> ( E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) )
37		inteq 3769	. . . . . . . . . . . 12 |- ( A = <. x , U. ran { A } >. -> |^| A = |^| <. x , U. ran { A } >. )
38	37	inteqd 3771	. . . . . . . . . . 11 |- ( A = <. x , U. ran { A } >. -> |^| |^| A = |^| |^| <. x , U. ran { A } >. )
39	38	adantl 275	. . . . . . . . . 10 |- ( ( A e. _V /\ A = <. x , U. ran { A } >. ) -> |^| |^| A = |^| |^| <. x , U. ran { A } >. )
40		op1stbg 4395	. . . . . . . . . . . 12 |- ( ( x e. _V /\ U. ran { A } e. _V ) -> |^| |^| <. x , U. ran { A } >. = x )
41	15, 28, 40	sylancr 410	. . . . . . . . . . 11 |- ( A e. _V -> |^| |^| <. x , U. ran { A } >. = x )
42	41	adantr 274	. . . . . . . . . 10 |- ( ( A e. _V /\ A = <. x , U. ran { A } >. ) -> |^| |^| <. x , U. ran { A } >. = x )
43	39, 42	eqtr2d 2171	. . . . . . . . 9 |- ( ( A e. _V /\ A = <. x , U. ran { A } >. ) -> x = |^| |^| A )
44	43	ex 114	. . . . . . . 8 |- ( A e. _V -> ( A = <. x , U. ran { A } >. -> x = |^| |^| A ) )
45	44	pm4.71rd 391	. . . . . . 7 |- ( A e. _V -> ( A = <. x , U. ran { A } >. <-> ( x = |^| |^| A /\ A = <. x , U. ran { A } >. ) ) )
46	45	anbi1d 460	. . . . . 6 |- ( A e. _V -> ( ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) <-> ( ( x = |^| |^| A /\ A = <. x , U. ran { A } >. ) /\ ( x e. B /\ U. ran { A } e. C ) ) ) )
47		anass 398	. . . . . . 7 |- ( ( ( x = |^| |^| A /\ A = <. x , U. ran { A } >. ) /\ ( x e. B /\ U. ran { A } e. C ) ) <-> ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) )
48	47	a1i 9	. . . . . 6 |- ( A e. _V -> ( ( ( x = |^| |^| A /\ A = <. x , U. ran { A } >. ) /\ ( x e. B /\ U. ran { A } e. C ) ) <-> ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) ) )
49	36, 46, 48	3bitrd 213	. . . . 5 |- ( A e. _V -> ( E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) ) )
50	49	exbidv 1797	. . . 4 |- ( A e. _V -> ( E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> E. x ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) ) )
51	11, 50	syl5bb 191	. . 3 |- ( A e. _V -> ( A e. ( B X. C ) <-> E. x ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) ) )
52		eqvisset 2691	. . . . . 6 |- ( x = |^| |^| A -> |^| |^| A e. _V )
53	52	adantr 274	. . . . 5 |- ( ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) -> |^| |^| A e. _V )
54	53	exlimiv 1577	. . . 4 |- ( E. x ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) -> |^| |^| A e. _V )
55	2	ad2antrl 481	. . . 4 |- ( ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) -> |^| |^| A e. _V )
56		opeq1 3700	. . . . . . 7 |- ( x = |^| |^| A -> <. x , U. ran { A } >. = <. |^| |^| A , U. ran { A } >. )
57	56	eqeq2d 2149	. . . . . 6 |- ( x = |^| |^| A -> ( A = <. x , U. ran { A } >. <-> A = <. |^| |^| A , U. ran { A } >. ) )
58		eleq1 2200	. . . . . . 7 |- ( x = |^| |^| A -> ( x e. B <-> |^| |^| A e. B ) )
59	58	anbi1d 460	. . . . . 6 |- ( x = |^| |^| A -> ( ( x e. B /\ U. ran { A } e. C ) <-> ( |^| |^| A e. B /\ U. ran { A } e. C ) ) )
60	57, 59	anbi12d 464	. . . . 5 |- ( x = |^| |^| A -> ( ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) <-> ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) ) )
61	60	ceqsexgv 2809	. . . 4 |- ( |^| |^| A e. _V -> ( E. x ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) <-> ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) ) )
62	54, 55, 61	pm5.21nii 693	. . 3 |- ( E. x ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) <-> ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) )
63	51, 62	syl6bb 195	. 2 |- ( A e. _V -> ( A e. ( B X. C ) <-> ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) ) )
64	1, 10, 63	pm5.21nii 693	1 |- ( A e. ( B X. C ) <-> ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   _Vcvv 2681   {csn 3522   <.cop 3525   U.cuni 3731   |^|cint 3766   X. cxp 4532   ran crn 4535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-dm 4544  df-rn 4545

This theorem is referenced by: (None)