Theorem xpopth 6144

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 6143	. . 3 |- ( A e. ( C X. D ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2		1st2nd2 6143	. . 3 |- ( B e. ( R X. S ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
3	1, 2	eqeqan12d 2181	. 2 |- ( ( A e. ( C X. D ) /\ B e. ( R X. S ) ) -> ( A = B <-> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` B ) , ( 2nd ` B ) >. ) )
4		1stexg 6135	. . . 4 |- ( A e. ( C X. D ) -> ( 1st ` A ) e. _V )
5		2ndexg 6136	. . . 4 |- ( A e. ( C X. D ) -> ( 2nd ` A ) e. _V )
6		opthg 4216	. . . 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 409	. . 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 274	. 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 188	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 103   <-> wb 104   = wceq 1343   e. wcel 2136   _Vcvv 2726   <.cop 3579   X. cxp 4602   ` cfv 5188   1stc1st 6106   2ndc2nd 6107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196  df-1st 6108  df-2nd 6109

This theorem is referenced by:  xmetxp  13147