Theorem xpopth 6322

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 6321	. . 3 |- ( A e. ( C X. D ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2		1st2nd2 6321	. . 3 |- ( B e. ( R X. S ) -> B = <. ( 1st ` B ) , ( 2nd ` B ) >. )
3	1, 2	eqeqan12d 2245	. 2 |- ( ( A e. ( C X. D ) /\ B e. ( R X. S ) ) -> ( A = B <-> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. ( 1st ` B ) , ( 2nd ` B ) >. ) )
4		1stexg 6313	. . . 4 |- ( A e. ( C X. D ) -> ( 1st ` A ) e. _V )
5		2ndexg 6314	. . . 4 |- ( A e. ( C X. D ) -> ( 2nd ` A ) e. _V )
6		opthg 4324	. . . 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 411	. . 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 276	. 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 189	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 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   <.cop 3669   X. cxp 4717   ` cfv 5318   1stc1st 6284   2ndc2nd 6285

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fo 5324  df-fv 5326  df-1st 6286  df-2nd 6287

This theorem is referenced by:  xmetxp  15181  wlkeq  16065