Theorem elxp6 6222

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5153. (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 2771	. 2 |- ( A e. ( B X. C ) -> A e. _V )
2		opexg 4257	. . . 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 2256	. . . 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 5153	. . 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 6195	. . . . . 6 |- ( A e. _V -> ( 1st ` A ) = U. dom { A } )
9		2ndvalg 6196	. . . . . 6 |- ( A e. _V -> ( 2nd ` A ) = U. ran { A } )
10	8, 9	opeq12d 3812	. . . . 5 |- ( A e. _V -> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. U. dom { A } , U. ran { A } >. )
11	10	eqeq2d 2205	. . . 4 |- ( A e. _V -> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. <-> A = <. U. dom { A } , U. ran { A } >. ) )
12	8	eleq1d 2262	. . . . 5 |- ( A e. _V -> ( ( 1st ` A ) e. B <-> U. dom { A } e. B ) )
13	9	eleq1d 2262	. . . . 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 2164   _Vcvv 2760   {csn 3618   <.cop 3621   U.cuni 3835   X. cxp 4657   dom cdm 4659   ran crn 4660   ` cfv 5254   1stc1st 6191   2ndc2nd 6192

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fv 5262  df-1st 6193  df-2nd 6194

This theorem is referenced by:  elxp7  6223  oprssdmm  6224  eqopi  6225  1st2nd2  6228  eldju2ndl  7131  eldju2ndr  7132  aptap  8669  qredeu  12235  qnumdencl  12325  tx1cn  14437  tx2cn  14438  psmetxrge0  14500  xmetxpbl  14676