Theorem elxp7 5817

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4828. (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 2610	. 2 |- ( A e. ( B X. C ) -> A e. _V )
2		elex 2610	. . 3 |- ( A e. ( _V X. _V ) -> A e. _V )
3	2	adantr 270	. 2 |- ( ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) -> A e. _V )
4		1stexg 5814	. . . . . . 7 |- ( A e. _V -> ( 1st ` A ) e. _V )
5		2ndexg 5815	. . . . . . 7 |- ( A e. _V -> ( 2nd ` A ) e. _V )
6	4, 5	jca 300	. . . . . 6 |- ( A e. _V -> ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) )
7	6	biantrud 298	. . . . 5 |- ( A e. _V -> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) ) ) )
8		elxp6 5816	. . . . 5 |- ( A e. ( _V X. _V ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) ) )
9	7, 8	syl6rbbr 197	. . . 4 |- ( A e. _V -> ( A e. ( _V X. _V ) <-> A = <. ( 1st ` A ) , ( 2nd ` A ) >. ) )
10	9	anbi1d 452	. . 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 ) ) ) )
11		elxp6 5816	. . 3 |- ( A e. ( B X. C ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
12	10, 11	syl6rbbr 197	. 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 652	1 |- ( A e. ( B X. C ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   _Vcvv 2601   <.cop 3401   X. cxp 4361   ` cfv 4922   1stc1st 5785   2ndc2nd 5786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928  df-fv 4930  df-1st 5787  df-2nd 5788

This theorem is referenced by:  xp2  5819  unielxp  5820  1stconst  5862  2ndconst  5863  f1od2  5876