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Theorem op1steq 5836
Description: Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.)
Assertion
Ref Expression
op1steq  |-  ( A  e.  ( V  X.  W )  ->  (
( 1st `  A
)  =  B  <->  E. x  A  =  <. B ,  x >. ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    V( x)    W( x)

Proof of Theorem op1steq
StepHypRef Expression
1 xpss 4474 . . 3  |-  ( V  X.  W )  C_  ( _V  X.  _V )
21sseli 2996 . 2  |-  ( A  e.  ( V  X.  W )  ->  A  e.  ( _V  X.  _V ) )
3 eqid 2082 . . . . . 6  |-  ( 2nd `  A )  =  ( 2nd `  A )
4 eqopi 5829 . . . . . 6  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  ( 2nd `  A
) ) )  ->  A  =  <. B , 
( 2nd `  A
) >. )
53, 4mpanr2 429 . . . . 5  |-  ( ( A  e.  ( _V 
X.  _V )  /\  ( 1st `  A )  =  B )  ->  A  =  <. B ,  ( 2nd `  A )
>. )
6 2ndexg 5826 . . . . . . 7  |-  ( A  e.  ( _V  X.  _V )  ->  ( 2nd `  A )  e.  _V )
7 opeq2 3579 . . . . . . . . 9  |-  ( x  =  ( 2nd `  A
)  ->  <. B ,  x >.  =  <. B , 
( 2nd `  A
) >. )
87eqeq2d 2093 . . . . . . . 8  |-  ( x  =  ( 2nd `  A
)  ->  ( A  =  <. B ,  x >.  <-> 
A  =  <. B , 
( 2nd `  A
) >. ) )
98spcegv 2687 . . . . . . 7  |-  ( ( 2nd `  A )  e.  _V  ->  ( A  =  <. B , 
( 2nd `  A
) >.  ->  E. x  A  =  <. B ,  x >. ) )
106, 9syl 14 . . . . . 6  |-  ( A  e.  ( _V  X.  _V )  ->  ( A  =  <. B ,  ( 2nd `  A )
>.  ->  E. x  A  = 
<. B ,  x >. ) )
1110adantr 270 . . . . 5  |-  ( ( A  e.  ( _V 
X.  _V )  /\  ( 1st `  A )  =  B )  ->  ( A  =  <. B , 
( 2nd `  A
) >.  ->  E. x  A  =  <. B ,  x >. ) )
125, 11mpd 13 . . . 4  |-  ( ( A  e.  ( _V 
X.  _V )  /\  ( 1st `  A )  =  B )  ->  E. x  A  =  <. B ,  x >. )
1312ex 113 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  ( ( 1st `  A )  =  B  ->  E. x  A  =  <. B ,  x >. ) )
14 eqop 5834 . . . . 5  |-  ( A  e.  ( _V  X.  _V )  ->  ( A  =  <. B ,  x >.  <-> 
( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  x ) ) )
15 simpl 107 . . . . 5  |-  ( ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  x )  -> 
( 1st `  A
)  =  B )
1614, 15syl6bi 161 . . . 4  |-  ( A  e.  ( _V  X.  _V )  ->  ( A  =  <. B ,  x >.  ->  ( 1st `  A
)  =  B ) )
1716exlimdv 1741 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  ( E. x  A  =  <. B ,  x >.  ->  ( 1st `  A )  =  B ) )
1813, 17impbid 127 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  ( ( 1st `  A )  =  B  <->  E. x  A  =  <. B ,  x >. ) )
192, 18syl 14 1  |-  ( A  e.  ( V  X.  W )  ->  (
( 1st `  A
)  =  B  <->  E. x  A  =  <. B ,  x >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   _Vcvv 2602   <.cop 3409    X. cxp 4369   ` cfv 4932   1stc1st 5796   2ndc2nd 5797
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fo 4938  df-fv 4940  df-1st 5798  df-2nd 5799
This theorem is referenced by:  releldm2  5842
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