Theorem elxp7 6138

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5091. (Contributed by NM, 19-Aug-2006.)

Assertion

Ref	Expression
elxp7	|- ( A e. ( B X. C ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )

Proof of Theorem elxp7

Step	Hyp	Ref	Expression
1		elex 2737	. 2 |- ( A e. ( B X. C ) -> A e. _V )
2		elex 2737	. . 3 |- ( A e. ( _V X. _V ) -> A e. _V )
3	2	adantr 274	. 2 |- ( ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) -> A e. _V )
4		elxp6 6137	. . 3 |- ( A e. ( B X. C ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
5		elxp6 6137	. . . . 5 |- ( A e. ( _V X. _V ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) ) )
6		1stexg 6135	. . . . . . 7 |- ( A e. _V -> ( 1st ` A ) e. _V )
7		2ndexg 6136	. . . . . . 7 |- ( A e. _V -> ( 2nd ` A ) e. _V )
8	6, 7	jca 304	. . . . . 6 |- ( A e. _V -> ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) )
9	8	biantrud 302	. . . . 5 |- ( A e. _V -> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) ) ) )
10	5, 9	bitr4id 198	. . . 4 |- ( A e. _V -> ( A e. ( _V X. _V ) <-> A = <. ( 1st ` A ) , ( 2nd ` A ) >. ) )
11	10	anbi1d 461	. . 3 |- ( A e. _V -> ( ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) ) )
12	4, 11	bitr4id 198	. 2 |- ( A e. _V -> ( A e. ( B X. C ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) ) )
13	1, 3, 12	pm5.21nii 694	1 |- ( A e. ( B X. C ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   <.cop 3579   X. cxp 4602   ` cfv 5188   1stc1st 6106   2ndc2nd 6107

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196  df-1st 6108  df-2nd 6109

This theorem is referenced by:  xp2  6141  unielxp  6142  1stconst  6189  2ndconst  6190  f1od2  6203