Theorem elxp7 6256

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5170. (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 2783	. 2 |- ( A e. ( B X. C ) -> A e. _V )
2		elex 2783	. . 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 6255	. . 3 |- ( A e. ( B X. C ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
5		elxp6 6255	. . . . 5 |- ( A e. ( _V X. _V ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) ) )
6		1stexg 6253	. . . . . . 7 |- ( A e. _V -> ( 1st ` A ) e. _V )
7		2ndexg 6254	. . . . . . 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 706	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 1373   e. wcel 2176   _Vcvv 2772   <.cop 3636   X. cxp 4673   ` cfv 5271   1stc1st 6224   2ndc2nd 6225

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6226  df-2nd 6227

This theorem is referenced by:  xp2  6259  unielxp  6260  1stconst  6307  2ndconst  6308  f1od2  6321