Theorem xpopth 6067

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 6066	. . 3 |- ( A e. ( C X. D ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2		1st2nd2 6066	. . 3 |- ( B e. ( R X. S ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
3	1, 2	eqeqan12d 2153	. 2 |- ( ( A e. ( C X. D ) /\ B e. ( R X. S ) ) -> ( A = B <-> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` B ) , ( 2nd ` B ) >. ) )
4		1stexg 6058	. . . 4 |- ( A e. ( C X. D ) -> ( 1st ` A ) e. _V )
5		2ndexg 6059	. . . 4 |- ( A e. ( C X. D ) -> ( 2nd ` A ) e. _V )
6		opthg 4155	. . . 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 408	. . 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 1331   e. wcel 1480   _Vcvv 2681   <.cop 3525   X. cxp 4532   ` cfv 5118   1stc1st 6029   2ndc2nd 6030

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126  df-1st 6031  df-2nd 6032

This theorem is referenced by:  xmetxp  12665