Theorem eqop 6323

Description: Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
eqop	|- ( A e. ( V X. W ) -> ( A = <. B , C >. <-> ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) )

Proof of Theorem eqop

Step	Hyp	Ref	Expression
1		1st2nd2 6321	. . 3 |- ( A e. ( V X. W ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2	1	eqeq1d 2238	. 2 |- ( A e. ( V X. W ) -> ( A = <. B , C >. <-> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. B , C >. ) )
3		1stexg 6313	. . 3 |- ( A e. ( V X. W ) -> ( 1st ` A ) e. _V )
4		2ndexg 6314	. . 3 |- ( A e. ( V X. W ) -> ( 2nd ` A ) e. _V )
5		opthg 4324	. . 3 |- ( ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. = <. B , C >. <-> ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) )
6	3, 4, 5	syl2anc 411	. 2 |- ( A e. ( V X. W ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. = <. B , C >. <-> ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) )
7	2, 6	bitrd 188	1 |- ( A e. ( V X. W ) -> ( A = <. B , C >. <-> ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) )

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:  eqop2  6324  op1steq  6325  f1od2  6381  txhmeo  14993