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Theorem op1steq 6383
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 4974 . . 3  |-  ( V  X.  W )  C_  ( _V  X.  _V )
21sseli 3336 . 2  |-  ( A  e.  ( V  X.  W )  ->  A  e.  ( _V  X.  _V ) )
3 eqid 2435 . . . . . 6  |-  ( 2nd `  A )  =  ( 2nd `  A )
4 eqopi 6375 . . . . . 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 5734 . . . . . 6  |-  ( 2nd `  A )  e.  _V
7 opeq2 3977 . . . . . . 7  |-  ( x  =  ( 2nd `  A
)  ->  <. B ,  x >.  =  <. B , 
( 2nd `  A
) >. )
87eqeq2d 2446 . . . . . 6  |-  ( x  =  ( 2nd `  A
)  ->  ( A  =  <. B ,  x >.  <-> 
A  =  <. B , 
( 2nd `  A
) >. ) )
96, 8spcev 3035 . . . . 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 6381 . . . . 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 1646 . . 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 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809    X. cxp 4868   ` cfv 5446   1stc1st 6339   2ndc2nd 6340
This theorem is referenced by:  releldm2  6389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-1st 6341  df-2nd 6342
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