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Theorem op1steq 6237
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 4771 . . 3  |-  ( V  X.  W )  C_  ( _V  X.  _V )
21sseli 3179 . 2  |-  ( A  e.  ( V  X.  W )  ->  A  e.  ( _V  X.  _V ) )
3 eqid 2196 . . . . . 6  |-  ( 2nd `  A )  =  ( 2nd `  A )
4 eqopi 6230 . . . . . 6  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  ( 2nd `  A
) ) )  ->  A  =  <. B , 
( 2nd `  A
) >. )
53, 4mpanr2 438 . . . . 5  |-  ( ( A  e.  ( _V 
X.  _V )  /\  ( 1st `  A )  =  B )  ->  A  =  <. B ,  ( 2nd `  A )
>. )
6 2ndexg 6226 . . . . . . 7  |-  ( A  e.  ( _V  X.  _V )  ->  ( 2nd `  A )  e.  _V )
7 opeq2 3809 . . . . . . . . 9  |-  ( x  =  ( 2nd `  A
)  ->  <. B ,  x >.  =  <. B , 
( 2nd `  A
) >. )
87eqeq2d 2208 . . . . . . . 8  |-  ( x  =  ( 2nd `  A
)  ->  ( A  =  <. B ,  x >.  <-> 
A  =  <. B , 
( 2nd `  A
) >. ) )
98spcegv 2852 . . . . . . 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 276 . . . . 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 115 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  ( ( 1st `  A )  =  B  ->  E. x  A  =  <. B ,  x >. ) )
14 eqop 6235 . . . . 5  |-  ( A  e.  ( _V  X.  _V )  ->  ( A  =  <. B ,  x >.  <-> 
( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  x ) ) )
15 simpl 109 . . . . 5  |-  ( ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  x )  -> 
( 1st `  A
)  =  B )
1614, 15biimtrdi 163 . . . 4  |-  ( A  e.  ( _V  X.  _V )  ->  ( A  =  <. B ,  x >.  ->  ( 1st `  A
)  =  B ) )
1716exlimdv 1833 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  ( E. x  A  =  <. B ,  x >.  ->  ( 1st `  A )  =  B ) )
1813, 17impbid 129 . 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 104    <-> wb 105    = wceq 1364   E.wex 1506    e. wcel 2167   _Vcvv 2763   <.cop 3625    X. cxp 4661   ` cfv 5258   1stc1st 6196   2ndc2nd 6197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fo 5264  df-fv 5266  df-1st 6198  df-2nd 6199
This theorem is referenced by:  releldm2  6243
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