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Theorem eqopi 6723
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 5057 . . 3  |-  ( V  X.  W )  C_  ( _V  X.  _V )
21sseli 3463 . 2  |-  ( A  e.  ( V  X.  W )  ->  A  e.  ( _V  X.  _V ) )
3 elxp6 6721 . . . 4  |-  ( A  e.  ( _V  X.  _V )  <->  ( A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) ) )
43simplbi 460 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
5 opeq12 4172 . . 3  |-  ( ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C )  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. B ,  C >. )
64, 5sylan9eq 2515 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) )  ->  A  =  <. B ,  C >. )
72, 6sylan 471 1  |-  ( ( A  e.  ( V  X.  W )  /\  ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) )  ->  A  =  <. B ,  C >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   <.cop 3994    X. cxp 4949   ` cfv 5529   1stc1st 6688   2ndc2nd 6689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fv 5537  df-1st 6690  df-2nd 6691
This theorem is referenced by:  op1steq  6731  dfoprab3  6743  1stconst  6774  2ndconst  6775  upxp  19338  cnvoprab  26201  el2xptp0  30298
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