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Theorem eqopi 6063
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 4642 . . 3  |-  ( V  X.  W )  C_  ( _V  X.  _V )
21sseli 3088 . 2  |-  ( A  e.  ( V  X.  W )  ->  A  e.  ( _V  X.  _V ) )
3 elxp6 6060 . . . 4  |-  ( A  e.  ( _V  X.  _V )  <->  ( A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) ) )
43simplbi 272 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
5 opeq12 3702 . . 3  |-  ( ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C )  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. B ,  C >. )
64, 5sylan9eq 2190 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) )  ->  A  =  <. B ,  C >. )
72, 6sylan 281 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 103    = wceq 1331    e. wcel 1480   _Vcvv 2681   <.cop 3525    X. cxp 4532   ` cfv 5118   1stc1st 6029   2ndc2nd 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fv 5126  df-1st 6031  df-2nd 6032
This theorem is referenced by:  op1steq  6070  dfoprab3  6082  1stconst  6111  2ndconst  6112  cnvoprab  6124  upxp  12430
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