Theorem eqop 6177

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 6175	. . 3 |- ( A e. ( V X. W ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2	1	eqeq1d 2186	. 2 |- ( A e. ( V X. W ) -> ( A = <. B , C >. <-> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. B , C >. ) )
3		1stexg 6167	. . 3 |- ( A e. ( V X. W ) -> ( 1st ` A ) e. _V )
4		2ndexg 6168	. . 3 |- ( A e. ( V X. W ) -> ( 2nd ` A ) e. _V )
5		opthg 4238	. . 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 1353   e. wcel 2148   _Vcvv 2737   <.cop 3595   X. cxp 4624   ` cfv 5216   1stc1st 6138   2ndc2nd 6139

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fo 5222  df-fv 5224  df-1st 6140  df-2nd 6141

This theorem is referenced by:  eqop2  6178  op1steq  6179  f1od2  6235  txhmeo  13750