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Theorem eqopi 6156
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 4793 . . 3  |-  ( V  X.  W )  C_  ( _V  X.  _V )
21sseli 3176 . 2  |-  ( A  e.  ( V  X.  W )  ->  A  e.  ( _V  X.  _V ) )
3 elxp6 6151 . . . 4  |-  ( A  e.  ( _V  X.  _V )  <->  ( A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) ) )
43simplbi 446 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
5 opeq12 3798 . . 3  |-  ( ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C )  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. B ,  C >. )
64, 5sylan9eq 2335 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) )  ->  A  =  <. B ,  C >. )
72, 6sylan 457 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 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    X. cxp 4687   ` cfv 5255   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  op1steq  6164  dfoprab3  6176  1stconst  6207  2ndconst  6208  upxp  17317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-1st 6122  df-2nd 6123
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