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Theorem eqop 6075
Description: Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
eqop  |-  ( A  e.  ( V  X.  W )  ->  ( A  =  <. B ,  C >. 
<->  ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) ) )

Proof of Theorem eqop
StepHypRef Expression
1 1st2nd2 6073 . . 3  |-  ( A  e.  ( V  X.  W )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
21eqeq1d 2148 . 2  |-  ( A  e.  ( V  X.  W )  ->  ( A  =  <. B ,  C >. 
<-> 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. B ,  C >. )
)
3 1stexg 6065 . . 3  |-  ( A  e.  ( V  X.  W )  ->  ( 1st `  A )  e. 
_V )
4 2ndexg 6066 . . 3  |-  ( A  e.  ( V  X.  W )  ->  ( 2nd `  A )  e. 
_V )
5 opthg 4160 . . 3  |-  ( ( ( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V )  -> 
( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. B ,  C >.  <->  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C ) ) )
63, 4, 5syl2anc 408 . 2  |-  ( A  e.  ( V  X.  W )  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. B ,  C >.  <->  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C ) ) )
72, 6bitrd 187 1  |-  ( A  e.  ( V  X.  W )  ->  ( A  =  <. B ,  C >. 
<->  ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   _Vcvv 2686   <.cop 3530    X. cxp 4537   ` cfv 5123   1stc1st 6036   2ndc2nd 6037
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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131  df-1st 6038  df-2nd 6039
This theorem is referenced by:  eqop2  6076  op1steq  6077  f1od2  6132  txhmeo  12504
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