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Theorem eqop2 5964
Description: Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)
Hypotheses
Ref Expression
eqop2.1  |-  B  e. 
_V
eqop2.2  |-  C  e. 
_V
Assertion
Ref Expression
eqop2  |-  ( A  =  <. B ,  C >.  <-> 
( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) ) )

Proof of Theorem eqop2
StepHypRef Expression
1 eqop2.1 . . . 4  |-  B  e. 
_V
2 eqop2.2 . . . 4  |-  C  e. 
_V
31, 2opelvv 4503 . . 3  |-  <. B ,  C >.  e.  ( _V 
X.  _V )
4 eleq1 2151 . . 3  |-  ( A  =  <. B ,  C >.  ->  ( A  e.  ( _V  X.  _V ) 
<-> 
<. B ,  C >.  e.  ( _V  X.  _V ) ) )
53, 4mpbiri 167 . 2  |-  ( A  =  <. B ,  C >.  ->  A  e.  ( _V  X.  _V )
)
6 eqop 5963 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  ( A  =  <. B ,  C >.  <-> 
( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) ) )
75, 6biadan2 445 1  |-  ( A  =  <. B ,  C >.  <-> 
( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1290    e. wcel 1439   _Vcvv 2622   <.cop 3455    X. cxp 4452   ` cfv 5030   1stc1st 5925   2ndc2nd 5926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-sbc 2844  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-mpt 3909  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-fo 5036  df-fv 5038  df-1st 5927  df-2nd 5928
This theorem is referenced by: (None)
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