Theorem eqop2 6416

Description: Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)

Hypotheses

Ref	Expression
eqop2.1	|- B e. _V
eqop2.2	|- C e. _V

Assertion

Ref	Expression
eqop2	|- ( A = <. B , C >. <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) )

Proof of Theorem eqop2

Step	Hyp	Ref	Expression
1		eqop2.1	. . . 4 |- B e. _V
2		eqop2.2	. . . 4 |- C e. _V
3	1, 2	opelvv 4827	. . 3 |- <. B , C >. e. ( _V X. _V )
4		eleq1 2301	. . 3 |- ( A = <. B , C >. -> ( A e. ( _V X. _V ) <-> <. B , C >. e. ( _V X. _V ) ) )
5	3, 4	mpbiri 168	. 2 |- ( A = <. B , C >. -> A e. ( _V X. _V ) )
6		eqop 6415	. 2 |- ( A e. ( _V X. _V ) -> ( A = <. B , C >. <-> ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) )
7	5, 6	biadan2 460	1 |- ( A = <. B , C >. <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) )

This proof depends on syntax axioms:   /\ 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: (None)