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Theorem eqopi 5942
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
StepHypRef Expression
1 xpss 4546 . . 3  |-  ( V  X.  W )  C_  ( _V  X.  _V )
21sseli 3021 . 2  |-  ( A  e.  ( V  X.  W )  ->  A  e.  ( _V  X.  _V ) )
3 elxp6 5940 . . . 4  |-  ( A  e.  ( _V  X.  _V )  <->  ( A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) ) )
43simplbi 268 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
5 opeq12 3624 . . 3  |-  ( ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C )  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. B ,  C >. )
64, 5sylan9eq 2140 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) )  ->  A  =  <. B ,  C >. )
72, 6sylan 277 1  |-  ( ( A  e.  ( V  X.  W )  /\  ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) )  ->  A  =  <. B ,  C >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   _Vcvv 2619   <.cop 3449    X. cxp 4436   ` cfv 5015   1stc1st 5909   2ndc2nd 5910
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-1st 5911  df-2nd 5912
This theorem is referenced by:  op1steq  5949  dfoprab3  5961  1stconst  5986  2ndconst  5987  cnvoprab  5999
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