Theorem elxp5 7111

Description: Membership in a Cartesian product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 7110 when the double intersection does not create class existence problems (caused by int0 4490). (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		elxp 5131	. 2 |- ( A e. ( B X. C ) <-> E. x E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) )
2		sneq 4187	. . . . . . . . . . . 12 |- ( A = <. x , y >. -> { A } = { <. x , y >. } )
3	2	rneqd 5353	. . . . . . . . . . 11 |- ( A = <. x , y >. -> ran { A } = ran { <. x , y >. } )
4	3	unieqd 4446	. . . . . . . . . 10 |- ( A = <. x , y >. -> U. ran { A } = U. ran { <. x , y >. } )
5		vex 3203	. . . . . . . . . . 11 |- x e. _V
6		vex 3203	. . . . . . . . . . 11 |- y e. _V
7	5, 6	op2nda 5620	. . . . . . . . . 10 |- U. ran { <. x , y >. } = y
8	4, 7	syl6req 2673	. . . . . . . . 9 |- ( A = <. x , y >. -> y = U. ran { A } )
9	8	pm4.71ri 665	. . . . . . . 8 |- ( A = <. x , y >. <-> ( y = U. ran { A } /\ A = <. x , y >. ) )
10	9	anbi1i 731	. . . . . . 7 |- ( ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( ( y = U. ran { A } /\ A = <. x , y >. ) /\ ( x e. B /\ y e. C ) ) )
11		anass 681	. . . . . . 7 |- ( ( ( 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 ) ) ) )
12	10, 11	bitri 264	. . . . . 6 |- ( ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( y = U. ran { A } /\ ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) ) )
13	12	exbii 1774	. . . . 5 |- ( 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 ) ) ) )
14		snex 4908	. . . . . . . 8 |- { A } e. _V
15	14	rnex 7100	. . . . . . 7 |- ran { A } e. _V
16	15	uniex 6953	. . . . . 6 |- U. ran { A } e. _V
17		opeq2 4403	. . . . . . . 8 |- ( y = U. ran { A } -> <. x , y >. = <. x , U. ran { A } >. )
18	17	eqeq2d 2632	. . . . . . 7 |- ( y = U. ran { A } -> ( A = <. x , y >. <-> A = <. x , U. ran { A } >. ) )
19		eleq1 2689	. . . . . . . 8 |- ( y = U. ran { A } -> ( y e. C <-> U. ran { A } e. C ) )
20	19	anbi2d 740	. . . . . . 7 |- ( y = U. ran { A } -> ( ( x e. B /\ y e. C ) <-> ( x e. B /\ U. ran { A } e. C ) ) )
21	18, 20	anbi12d 747	. . . . . 6 |- ( 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 ) ) ) )
22	16, 21	ceqsexv 3242	. . . . 5 |- ( 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 ) ) )
23	13, 22	bitri 264	. . . 4 |- ( E. y ( A = <. x , y >. /\ ( x e. B /\ y e. C ) ) <-> ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) )
24		inteq 4478	. . . . . . . 8 |- ( A = <. x , U. ran { A } >. -> |^| A = |^| <. x , U. ran { A } >. )
25	24	inteqd 4480	. . . . . . 7 |- ( A = <. x , U. ran { A } >. -> |^| |^| A = |^| |^| <. x , U. ran { A } >. )
26	5, 16	op1stb 4940	. . . . . . 7 |- |^| |^| <. x , U. ran { A } >. = x
27	25, 26	syl6req 2673	. . . . . 6 |- ( A = <. x , U. ran { A } >. -> x = |^| |^| A )
28	27	pm4.71ri 665	. . . . 5 |- ( A = <. x , U. ran { A } >. <-> ( x = |^| |^| A /\ A = <. x , U. ran { A } >. ) )
29	28	anbi1i 731	. . . 4 |- ( ( 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 ) ) )
30		anass 681	. . . 4 |- ( ( ( 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 ) ) ) )
31	23, 29, 30	3bitri 286	. . 3 |- ( 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 ) ) ) )
32	31	exbii 1774	. 2 |- ( 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 ) ) ) )
33		eqvisset 3211	. . . . 5 |- ( x = |^| |^| A -> |^| |^| A e. _V )
34	33	adantr 481	. . . 4 |- ( ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) -> |^| |^| A e. _V )
35	34	exlimiv 1858	. . 3 |- ( E. x ( x = |^| |^| A /\ ( A = <. x , U. ran { A } >. /\ ( x e. B /\ U. ran { A } e. C ) ) ) -> |^| |^| A e. _V )
36		elex 3212	. . . 4 |- ( |^| |^| A e. B -> |^| |^| A e. _V )
37	36	ad2antrl 764	. . 3 |- ( ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) -> |^| |^| A e. _V )
38		opeq1 4402	. . . . . 6 |- ( x = |^| |^| A -> <. x , U. ran { A } >. = <. |^| |^| A , U. ran { A } >. )
39	38	eqeq2d 2632	. . . . 5 |- ( x = |^| |^| A -> ( A = <. x , U. ran { A } >. <-> A = <. |^| |^| A , U. ran { A } >. ) )
40		eleq1 2689	. . . . . 6 |- ( x = |^| |^| A -> ( x e. B <-> |^| |^| A e. B ) )
41	40	anbi1d 741	. . . . 5 |- ( x = |^| |^| A -> ( ( x e. B /\ U. ran { A } e. C ) <-> ( |^| |^| A e. B /\ U. ran { A } e. C ) ) )
42	39, 41	anbi12d 747	. . . 4 |- ( 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 ) ) ) )
43	42	ceqsexgv 3335	. . 3 |- ( |^| |^| 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 ) ) ) )
44	35, 37, 43	pm5.21nii 368	. 2 |- ( 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 ) ) )
45	1, 32, 44	3bitri 286	1 |- ( A e. ( B X. C ) <-> ( A = <. |^| |^| A , U. ran { A } >. /\ ( |^| |^| A e. B /\ U. ran { A } e. C ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   U.cuni 4436   |^|cint 4475   X. cxp 5112   ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by: (None)