Theorem eqopi 5829

Description: Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)

Assertion

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

Proof of Theorem eqopi

Step	Hyp	Ref	Expression
1		xpss 4474	. . 3 |- ( V X. W ) C_ ( _V X. _V )
2	1	sseli 2996	. 2 |- ( A e. ( V X. W ) -> A e. ( _V X. _V ) )
3		elxp6 5827	. . . 4 |- ( A e. ( _V X. _V ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) ) )
4	3	simplbi 268	. . 3 |- ( A e. ( _V X. _V ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
5		opeq12 3580	. . 3 |- ( ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. B , C >. )
6	4, 5	sylan9eq 2134	. 2 |- ( ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) -> A = <. B , C >. )
7	2, 6	sylan 277	1 |- ( ( A e. ( V X. W ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) -> A = <. B , C >. )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   _Vcvv 2602   <.cop 3409   X. cxp 4369   ` cfv 4932   1stc1st 5796   2ndc2nd 5797

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fv 4940  df-1st 5798  df-2nd 5799

This theorem is referenced by:  op1steq  5836  dfoprab3  5848  1stconst  5873  2ndconst  5874  cnvoprab  5886