Theorem elxp6 6227

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5157. (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 2774	. 2 |- ( A e. ( B X. C ) -> A e. _V )
2		opexg 4261	. . . 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 2259	. . . 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 5157	. . 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 6200	. . . . . 6 |- ( A e. _V -> ( 1st ` A ) = U. dom { A } )
9		2ndvalg 6201	. . . . . 6 |- ( A e. _V -> ( 2nd ` A ) = U. ran { A } )
10	8, 9	opeq12d 3816	. . . . 5 |- ( A e. _V -> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. U. dom { A } , U. ran { A } >. )
11	10	eqeq2d 2208	. . . 4 |- ( A e. _V -> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. <-> A = <. U. dom { A } , U. ran { A } >. ) )
12	8	eleq1d 2265	. . . . 5 |- ( A e. _V -> ( ( 1st ` A ) e. B <-> U. dom { A } e. B ) )
13	9	eleq1d 2265	. . . . 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 705	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 1364   e. wcel 2167   _Vcvv 2763   {csn 3622   <.cop 3625   U.cuni 3839   X. cxp 4661   dom cdm 4663   ran crn 4664   ` cfv 5258   1stc1st 6196   2ndc2nd 6197

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fv 5266  df-1st 6198  df-2nd 6199

This theorem is referenced by:  elxp7  6228  oprssdmm  6229  eqopi  6230  1st2nd2  6233  eldju2ndl  7138  eldju2ndr  7139  aptap  8677  qredeu  12265  qnumdencl  12355  tx1cn  14505  tx2cn  14506  psmetxrge0  14568  xmetxpbl  14744