Theorem eqop 6415

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 6413	. . 3 |- ( A e. ( V X. W ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2	1	eqeq1d 2247	. 2 |- ( A e. ( V X. W ) -> ( A = <. B , C >. <-> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. B , C >. ) )
3		1stexg 6405	. . 3 |- ( A e. ( V X. W ) -> ( 1st ` A ) e. _V )
4		2ndexg 6406	. . 3 |- ( A e. ( V X. W ) -> ( 2nd ` A ) e. _V )
5		opthg 4380	. . 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 415	. 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   <.cop 3712   X. cxp 4774   ` cfv 5380   1stc1st 6376   2ndc2nd 6377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fo 5386  df-fv 5388  df-1st 6378  df-2nd 6379

This theorem is used by:  eqop2  6416  op1steq  6417  f1od2  6475  txhmeo  15545