Theorem elxp7 6394

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5270. (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 2833	. 2 |- ( A e. ( B X. C ) -> A e. _V )
2		elex 2833	. . 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 6393	. . 3 |- ( A e. ( B X. C ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )
5		elxp6 6393	. . . . 5 |- ( A e. ( _V X. _V ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) ) )
6		1stexg 6391	. . . . . . 7 |- ( A e. _V -> ( 1st ` A ) e. _V )
7		2ndexg 6392	. . . . . . 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 469	. . 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 716	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 1402   e. wcel 2209   _Vcvv 2821   <.cop 3708   X. cxp 4767   ` cfv 5372   1stc1st 6362   2ndc2nd 6363

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-1st 6364  df-2nd 6365

This theorem is referenced by:  xp2  6397  unielxp  6398  1stconst  6447  2ndconst  6448  f1od2  6461