Theorem eqop2 6340

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 4776	. . 3 |- <. B , C >. e. ( _V X. _V )
4		eleq1 2294	. . 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 6339	. 2 |- ( A e. ( _V X. _V ) -> ( A = <. B , C >. <-> ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) )
7	5, 6	biadan2 456	1 |- ( A = <. B , C >. <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   _Vcvv 2802   <.cop 3672   X. cxp 4723   ` cfv 5326   1stc1st 6300   2ndc2nd 6301

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-1st 6302  df-2nd 6303

This theorem is referenced by: (None)