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Theorem op1steq 6617
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 4945 . . 3  |-  ( V  X.  W )  C_  ( _V  X.  _V )
21sseli 3351 . 2  |-  ( A  e.  ( V  X.  W )  ->  A  e.  ( _V  X.  _V ) )
3 eqid 2442 . . . . . 6  |-  ( 2nd `  A )  =  ( 2nd `  A )
4 eqopi 6609 . . . . . 6  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  ( 2nd `  A
) ) )  ->  A  =  <. B , 
( 2nd `  A
) >. )
53, 4mpanr2 684 . . . . 5  |-  ( ( A  e.  ( _V 
X.  _V )  /\  ( 1st `  A )  =  B )  ->  A  =  <. B ,  ( 2nd `  A )
>. )
6 fvex 5700 . . . . . 6  |-  ( 2nd `  A )  e.  _V
7 opeq2 4059 . . . . . . 7  |-  ( x  =  ( 2nd `  A
)  ->  <. B ,  x >.  =  <. B , 
( 2nd `  A
) >. )
87eqeq2d 2453 . . . . . 6  |-  ( x  =  ( 2nd `  A
)  ->  ( A  =  <. B ,  x >.  <-> 
A  =  <. B , 
( 2nd `  A
) >. ) )
96, 8spcev 3063 . . . . 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 434 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  ( ( 1st `  A )  =  B  ->  E. x  A  =  <. B ,  x >. ) )
12 eqop 6615 . . . . 5  |-  ( A  e.  ( _V  X.  _V )  ->  ( A  =  <. B ,  x >.  <-> 
( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  x ) ) )
13 simpl 457 . . . . 5  |-  ( ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  x )  -> 
( 1st `  A
)  =  B )
1412, 13syl6bi 228 . . . 4  |-  ( A  e.  ( _V  X.  _V )  ->  ( A  =  <. B ,  x >.  ->  ( 1st `  A
)  =  B ) )
1514exlimdv 1690 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  ( E. x  A  =  <. B ,  x >.  ->  ( 1st `  A )  =  B ) )
1611, 15impbid 191 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2971   <.cop 3882    X. cxp 4837   ` cfv 5417   1stc1st 6574   2ndc2nd 6575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5380  df-fun 5419  df-fv 5425  df-1st 6576  df-2nd 6577
This theorem is referenced by:  releldm2  6623
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