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Theorem eqop 5947
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 5945 . . 3  |-  ( A  e.  ( V  X.  W )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
21eqeq1d 2096 . 2  |-  ( A  e.  ( V  X.  W )  ->  ( A  =  <. B ,  C >. 
<-> 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. B ,  C >. )
)
3 1stexg 5938 . . 3  |-  ( A  e.  ( V  X.  W )  ->  ( 1st `  A )  e. 
_V )
4 2ndexg 5939 . . 3  |-  ( A  e.  ( V  X.  W )  ->  ( 2nd `  A )  e. 
_V )
5 opthg 4065 . . 3  |-  ( ( ( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V )  -> 
( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. B ,  C >.  <->  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C ) ) )
63, 4, 5syl2anc 403 . 2  |-  ( A  e.  ( V  X.  W )  ->  ( <. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. B ,  C >.  <->  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C ) ) )
72, 6bitrd 186 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   _Vcvv 2619   <.cop 3449    X. cxp 4436   ` cfv 5015   1stc1st 5909   2ndc2nd 5910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fo 5021  df-fv 5023  df-1st 5911  df-2nd 5912
This theorem is referenced by:  eqop2  5948  op1steq  5949  f1od2  6000
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