Theorem xpopth 5822

Description: An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.)

Assertion

Ref	Expression
xpopth	|- ( ( A e. ( C X. D ) /\ B e. ( R X. S ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> A = B ) )

Proof of Theorem xpopth

Step	Hyp	Ref	Expression
1		1st2nd2 5821	. . 3 |- ( A e. ( C X. D ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2		1st2nd2 5821	. . 3 |- ( B e. ( R X. S ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
3	1, 2	eqeqan12d 2096	. 2 |- ( ( A e. ( C X. D ) /\ B e. ( R X. S ) ) -> ( A = B <-> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` B ) , ( 2nd ` B ) >. ) )
4		1stexg 5814	. . . 4 |- ( A e. ( C X. D ) -> ( 1st ` A ) e. _V )
5		2ndexg 5815	. . . 4 |- ( A e. ( C X. D ) -> ( 2nd ` A ) e. _V )
6		opthg 3993	. . . 4 |- ( ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` B ) , ( 2nd ` B ) >. <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )
7	4, 5, 6	syl2anc 403	. . 3 |- ( A e. ( C X. D ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` B ) , ( 2nd ` B ) >. <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )
8	7	adantr 270	. 2 |- ( ( A e. ( C X. D ) /\ B e. ( R X. S ) ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` B ) , ( 2nd ` B ) >. <-> ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) ) )
9	3, 8	bitr2d 187	1 |- ( ( A e. ( C X. D ) /\ B e. ( R X. S ) ) -> ( ( ( 1st ` A ) = ( 1st ` B ) /\ ( 2nd ` A ) = ( 2nd ` B ) ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   _Vcvv 2601   <.cop 3401   X. cxp 4361   ` cfv 4922   1stc1st 5785   2ndc2nd 5786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928  df-fv 4930  df-1st 5787  df-2nd 5788

This theorem is referenced by: (None)