Theorem elxp6 6365

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5252. (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 2827	. 2 |- ( A e. ( B X. C ) -> A e. _V )
2		opexg 4346	. . . 4 |- ( ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. _V )
3	2	adantl 277	. . 3 |- ( ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. _V )
4		eleq1 2297	. . . 4 |- ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. -> ( A e. _V <-> <. ( 1st ` A ) , ( 2nd ` A ) >. e. _V ) )
5	4	adantr 276	. . 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 167	. 2 |- ( ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) -> A e. _V )
7		elxp4 5252	. . 3 |- ( A e. ( B X. C ) <-> ( A = <. U. dom { A } , U. ran { A } >. /\ ( U. dom { A } e. B /\ U. ran { A } e. C ) ) )
8		1stvalg 6338	. . . . . 6 |- ( A e. _V -> ( 1st ` A ) = U. dom { A } )
9		2ndvalg 6339	. . . . . 6 |- ( A e. _V -> ( 2nd ` A ) = U. ran { A } )
10	8, 9	opeq12d 3893	. . . . 5 |- ( A e. _V -> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. U. dom { A } , U. ran { A } >. )
11	10	eqeq2d 2246	. . . 4 |- ( A e. _V -> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. <-> A = <. U. dom { A } , U. ran { A } >. ) )
12	8	eleq1d 2303	. . . . 5 |- ( A e. _V -> ( ( 1st ` A ) e. B <-> U. dom { A } e. B ) )
13	9	eleq1d 2303	. . . . 5 |- ( A e. _V -> ( ( 2nd ` A ) e. C <-> U. ran { A } e. C ) )
14	12, 13	anbi12d 473	. . . 4 |- ( A e. _V -> ( ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) <-> ( U. dom { A } e. B /\ U. ran { A } e. C ) ) )
15	11, 14	anbi12d 473	. . 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 ) ) ) )
16	7, 15	bitr4id 199	. 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 712	1 |- ( A e. ( B X. C ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. B /\ ( 2nd ` A ) e. C ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   _Vcvv 2815   {csn 3691   <.cop 3694   U.cuni 3916   X. cxp 4749   dom cdm 4751   ran crn 4752   ` cfv 5354   1stc1st 6334   2ndc2nd 6335

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fv 5362  df-1st 6336  df-2nd 6337

This theorem is referenced by:  elxp7  6366  oprssdmm  6367  eqopi  6368  1st2nd2  6371  eldju2ndl  7365  eldju2ndr  7366  aptap  8926  qredeu  12798  qnumdencl  12888  tx1cn  15151  tx2cn  15152  psmetxrge0  15214  xmetxpbl  15390