Theorem elxp7 6185

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5128. (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 2760	. 2 |- ( A e. ( B X. C ) -> A e. _V )
2		elex 2760	. . 3 |- ( A e. ( _V X. _V ) -> A e. _V )
3	2	adantr 276	. 2 |- ( ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) -> A e. _V )
4		elxp6 6184	. . 3 |- ( A e. ( B X. C ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
5		elxp6 6184	. . . . 5 |- ( A e. ( _V X. _V ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) ) )
6		1stexg 6182	. . . . . . 7 |- ( A e. _V -> ( 1st ` A ) e. _V )
7		2ndexg 6183	. . . . . . 7 |- ( A e. _V -> ( 2nd ` A ) e. _V )
8	6, 7	jca 306	. . . . . 6 |- ( A e. _V -> ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) )
9	8	biantrud 304	. . . . 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 199	. . . 4 |- ( A e. _V -> ( A e. ( _V X. _V ) <-> A = <. ( 1st ` A ) , ( 2nd ` A ) >. ) )
11	10	anbi1d 465	. . 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 199	. 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 705	1 |- ( A e. ( B X. C ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1363   e. wcel 2158   _Vcvv 2749   <.cop 3607   X. cxp 4636   ` cfv 5228   1stc1st 6153   2ndc2nd 6154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fo 5234  df-fv 5236  df-1st 6155  df-2nd 6156

This theorem is referenced by:  xp2  6188  unielxp  6189  1stconst  6236  2ndconst  6237  f1od2  6250