Theorem elxp6 6315

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5216. (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 2811	. 2 |- ( A e. ( B X. C ) -> A e. _V )
2		opexg 4314	. . . 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 2292	. . . 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 5216	. . 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 6288	. . . . . 6 |- ( A e. _V -> ( 1st ` A ) = U. dom { A } )
9		2ndvalg 6289	. . . . . 6 |- ( A e. _V -> ( 2nd ` A ) = U. ran { A } )
10	8, 9	opeq12d 3865	. . . . 5 |- ( A e. _V -> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. U. dom { A } , U. ran { A } >. )
11	10	eqeq2d 2241	. . . 4 |- ( A e. _V -> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. <-> A = <. U. dom { A } , U. ran { A } >. ) )
12	8	eleq1d 2298	. . . . 5 |- ( A e. _V -> ( ( 1st ` A ) e. B <-> U. dom { A } e. B ) )
13	9	eleq1d 2298	. . . . 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 709	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 1395   e. wcel 2200   _Vcvv 2799   {csn 3666   <.cop 3669   U.cuni 3888   X. cxp 4717   dom cdm 4719   ran crn 4720   ` cfv 5318   1stc1st 6284   2ndc2nd 6285

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fv 5326  df-1st 6286  df-2nd 6287

This theorem is referenced by:  elxp7  6316  oprssdmm  6317  eqopi  6318  1st2nd2  6321  eldju2ndl  7239  eldju2ndr  7240  aptap  8797  qredeu  12619  qnumdencl  12709  tx1cn  14943  tx2cn  14944  psmetxrge0  15006  xmetxpbl  15182