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Theorem op1steq 6330
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 4922 . . 3  |-  ( V  X.  W )  C_  ( _V  X.  _V )
21sseli 3287 . 2  |-  ( A  e.  ( V  X.  W )  ->  A  e.  ( _V  X.  _V ) )
3 eqid 2387 . . . . . 6  |-  ( 2nd `  A )  =  ( 2nd `  A )
4 eqopi 6322 . . . . . 6  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  ( 2nd `  A
) ) )  ->  A  =  <. B , 
( 2nd `  A
) >. )
53, 4mpanr2 666 . . . . 5  |-  ( ( A  e.  ( _V 
X.  _V )  /\  ( 1st `  A )  =  B )  ->  A  =  <. B ,  ( 2nd `  A )
>. )
6 fvex 5682 . . . . . 6  |-  ( 2nd `  A )  e.  _V
7 opeq2 3927 . . . . . . 7  |-  ( x  =  ( 2nd `  A
)  ->  <. B ,  x >.  =  <. B , 
( 2nd `  A
) >. )
87eqeq2d 2398 . . . . . 6  |-  ( x  =  ( 2nd `  A
)  ->  ( A  =  <. B ,  x >.  <-> 
A  =  <. B , 
( 2nd `  A
) >. ) )
96, 8spcev 2986 . . . . 5  |-  ( A  =  <. B ,  ( 2nd `  A )
>.  ->  E. x  A  = 
<. B ,  x >. )
105, 9syl 16 . . . 4  |-  ( ( A  e.  ( _V 
X.  _V )  /\  ( 1st `  A )  =  B )  ->  E. x  A  =  <. B ,  x >. )
1110ex 424 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  ( ( 1st `  A )  =  B  ->  E. x  A  =  <. B ,  x >. ) )
12 eqop 6328 . . . . 5  |-  ( A  e.  ( _V  X.  _V )  ->  ( A  =  <. B ,  x >.  <-> 
( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  x ) ) )
13 simpl 444 . . . . 5  |-  ( ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  x )  -> 
( 1st `  A
)  =  B )
1412, 13syl6bi 220 . . . 4  |-  ( A  e.  ( _V  X.  _V )  ->  ( A  =  <. B ,  x >.  ->  ( 1st `  A
)  =  B ) )
1514exlimdv 1643 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  ( E. x  A  =  <. B ,  x >.  ->  ( 1st `  A )  =  B ) )
1611, 15impbid 184 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  ( ( 1st `  A )  =  B  <->  E. x  A  =  <. B ,  x >. ) )
172, 16syl 16 1  |-  ( A  e.  ( V  X.  W )  ->  (
( 1st `  A
)  =  B  <->  E. x  A  =  <. B ,  x >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2899   <.cop 3760    X. cxp 4816   ` cfv 5394   1stc1st 6286   2ndc2nd 6287
This theorem is referenced by:  releldm2  6336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fv 5402  df-1st 6288  df-2nd 6289
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