Theorem elxp6 5796

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4808. (Contributed by NM, 9-Oct-2004.)

Assertion

Ref	Expression
elxp6	|- ( A e. ( B X. C ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )

Proof of Theorem elxp6

Step	Hyp	Ref	Expression
1		elex 2566	. 2 |- ( A e. ( B X. C ) -> A e. _V )
2		opexg 3964	. . . 4 |- ( ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. _V )
3	2	adantl 262	. . 3 |- ( ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. _V )
4		eleq1 2100	. . . 4 |- ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. -> ( A e. _V <-> <. ( 1st ` A ) , ( 2nd ` A ) >. e. _V ) )
5	4	adantr 261	. . 3 |- ( ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) -> ( A e. _V <-> <. ( 1st ` A ) , ( 2nd ` A ) >. e. _V ) )
6	3, 5	mpbird 156	. 2 |- ( ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) -> A e. _V )
7		1stvalg 5769	. . . . . 6 |- ( A e. _V -> ( 1st ` A ) = U. dom { A } )
8		2ndvalg 5770	. . . . . 6 |- ( A e. _V -> ( 2nd ` A ) = U. ran { A } )
9	7, 8	opeq12d 3557	. . . . 5 |- ( A e. _V -> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. U. dom { A } , U. ran { A } >. )
10	9	eqeq2d 2051	. . . 4 |- ( A e. _V -> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. <-> A = <. U. dom { A } , U. ran { A } >. ) )
11	7	eleq1d 2106	. . . . 5 |- ( A e. _V -> ( ( 1st ` A ) e. B <-> U. dom { A } e. B ) )
12	8	eleq1d 2106	. . . . 5 |- ( A e. _V -> ( ( 2nd ` A ) e. C <-> U. ran { A } e. C ) )
13	11, 12	anbi12d 442	. . . 4 |- ( A e. _V -> ( ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) <-> ( U. dom { A } e. B /\ U. ran { A } e. C ) ) )
14	10, 13	anbi12d 442	. . 3 |- ( A e. _V -> ( ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) <-> ( A = <. U. dom { A } , U. ran { A } >. /\ ( U. dom { A } e. B /\ U. ran { A } e. C ) ) ) )
15		elxp4 4808	. . 3 |- ( A e. ( B X. C ) <-> ( A = <. U. dom { A } , U. ran { A } >. /\ ( U. dom { A } e. B /\ U. ran { A } e. C ) ) )
16	14, 15	syl6rbbr 188	. 2 |- ( A e. _V -> ( A e. ( B X. C ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) ) )
17	1, 6, 16	pm5.21nii 620	1 |- ( A e. ( B X. C ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   _Vcvv 2557   {csn 3375   <.cop 3378   U.cuni 3580   X. cxp 4343   dom cdm 4345   ran crn 4346   ` cfv 4902   1stc1st 5765   2ndc2nd 5766

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-1st 5767  df-2nd 5768

This theorem is referenced by:  elxp7  5797  eqopi  5798  1st2nd2  5801