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Theorem fgreu 27661
Description: Exactly one point of a function's graph has a given first element. (Contributed by Thierry Arnoux, 1-Apr-2018.)
Assertion
Ref Expression
fgreu  |-  ( ( Fun  F  /\  X  e.  dom  F )  ->  E! p  e.  F  X  =  ( 1st `  p ) )
Distinct variable groups:    F, p    X, p

Proof of Theorem fgreu
Dummy variable  q is distinct from all other variables.
StepHypRef Expression
1 funfvop 6000 . . 3  |-  ( ( Fun  F  /\  X  e.  dom  F )  ->  <. X ,  ( F `
 X ) >.  e.  F )
2 simplll 759 . . . . . . . 8  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  X  =  ( 1st `  p
) )  ->  Fun  F )
3 funrel 5611 . . . . . . . 8  |-  ( Fun 
F  ->  Rel  F )
42, 3syl 16 . . . . . . 7  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  X  =  ( 1st `  p
) )  ->  Rel  F )
5 simplr 755 . . . . . . 7  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  X  =  ( 1st `  p
) )  ->  p  e.  F )
6 1st2nd 6845 . . . . . . 7  |-  ( ( Rel  F  /\  p  e.  F )  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
74, 5, 6syl2anc 661 . . . . . 6  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  X  =  ( 1st `  p
) )  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
8 simpr 461 . . . . . . 7  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  X  =  ( 1st `  p
) )  ->  X  =  ( 1st `  p
) )
9 simpllr 760 . . . . . . . 8  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  X  =  ( 1st `  p
) )  ->  X  e.  dom  F )
108opeq1d 4225 . . . . . . . . . 10  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  X  =  ( 1st `  p
) )  ->  <. X , 
( 2nd `  p
) >.  =  <. ( 1st `  p ) ,  ( 2nd `  p
) >. )
117, 10eqtr4d 2501 . . . . . . . . 9  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  X  =  ( 1st `  p
) )  ->  p  =  <. X ,  ( 2nd `  p )
>. )
1211, 5eqeltrrd 2546 . . . . . . . 8  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  X  =  ( 1st `  p
) )  ->  <. X , 
( 2nd `  p
) >.  e.  F )
13 funopfvb 5916 . . . . . . . . 9  |-  ( ( Fun  F  /\  X  e.  dom  F )  -> 
( ( F `  X )  =  ( 2nd `  p )  <->  <. X ,  ( 2nd `  p ) >.  e.  F
) )
1413biimpar 485 . . . . . . . 8  |-  ( ( ( Fun  F  /\  X  e.  dom  F )  /\  <. X ,  ( 2nd `  p )
>.  e.  F )  -> 
( F `  X
)  =  ( 2nd `  p ) )
152, 9, 12, 14syl21anc 1227 . . . . . . 7  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  X  =  ( 1st `  p
) )  ->  ( F `  X )  =  ( 2nd `  p
) )
168, 15opeq12d 4227 . . . . . 6  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  X  =  ( 1st `  p
) )  ->  <. X , 
( F `  X
) >.  =  <. ( 1st `  p ) ,  ( 2nd `  p
) >. )
177, 16eqtr4d 2501 . . . . 5  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  X  =  ( 1st `  p
) )  ->  p  =  <. X ,  ( F `  X )
>. )
18 simpr 461 . . . . . . 7  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  p  =  <. X ,  ( F `  X )
>. )  ->  p  = 
<. X ,  ( F `
 X ) >.
)
1918fveq2d 5876 . . . . . 6  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  p  =  <. X ,  ( F `  X )
>. )  ->  ( 1st `  p )  =  ( 1st `  <. X , 
( F `  X
) >. ) )
20 fvex 5882 . . . . . . . 8  |-  ( F `
 X )  e. 
_V
21 op1stg 6811 . . . . . . . 8  |-  ( ( X  e.  dom  F  /\  ( F `  X
)  e.  _V )  ->  ( 1st `  <. X ,  ( F `  X ) >. )  =  X )
2220, 21mpan2 671 . . . . . . 7  |-  ( X  e.  dom  F  -> 
( 1st `  <. X ,  ( F `  X ) >. )  =  X )
2322ad3antlr 730 . . . . . 6  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  p  =  <. X ,  ( F `  X )
>. )  ->  ( 1st `  <. X ,  ( F `  X )
>. )  =  X
)
2419, 23eqtr2d 2499 . . . . 5  |-  ( ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F )  /\  p  =  <. X ,  ( F `  X )
>. )  ->  X  =  ( 1st `  p
) )
2517, 24impbida 832 . . . 4  |-  ( ( ( Fun  F  /\  X  e.  dom  F )  /\  p  e.  F
)  ->  ( X  =  ( 1st `  p
)  <->  p  =  <. X ,  ( F `  X ) >. )
)
2625ralrimiva 2871 . . 3  |-  ( ( Fun  F  /\  X  e.  dom  F )  ->  A. p  e.  F  ( X  =  ( 1st `  p )  <->  p  =  <. X ,  ( F `
 X ) >.
) )
27 eqeq2 2472 . . . . . 6  |-  ( q  =  <. X ,  ( F `  X )
>.  ->  ( p  =  q  <->  p  =  <. X ,  ( F `  X ) >. )
)
2827bibi2d 318 . . . . 5  |-  ( q  =  <. X ,  ( F `  X )
>.  ->  ( ( X  =  ( 1st `  p
)  <->  p  =  q
)  <->  ( X  =  ( 1st `  p
)  <->  p  =  <. X ,  ( F `  X ) >. )
) )
2928ralbidv 2896 . . . 4  |-  ( q  =  <. X ,  ( F `  X )
>.  ->  ( A. p  e.  F  ( X  =  ( 1st `  p
)  <->  p  =  q
)  <->  A. p  e.  F  ( X  =  ( 1st `  p )  <->  p  =  <. X ,  ( F `
 X ) >.
) ) )
3029rspcev 3210 . . 3  |-  ( (
<. X ,  ( F `
 X ) >.  e.  F  /\  A. p  e.  F  ( X  =  ( 1st `  p
)  <->  p  =  <. X ,  ( F `  X ) >. )
)  ->  E. q  e.  F  A. p  e.  F  ( X  =  ( 1st `  p
)  <->  p  =  q
) )
311, 26, 30syl2anc 661 . 2  |-  ( ( Fun  F  /\  X  e.  dom  F )  ->  E. q  e.  F  A. p  e.  F  ( X  =  ( 1st `  p )  <->  p  =  q ) )
32 reu6 3288 . 2  |-  ( E! p  e.  F  X  =  ( 1st `  p
)  <->  E. q  e.  F  A. p  e.  F  ( X  =  ( 1st `  p )  <->  p  =  q ) )
3331, 32sylibr 212 1  |-  ( ( Fun  F  /\  X  e.  dom  F )  ->  E! p  e.  F  X  =  ( 1st `  p ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   E!wreu 2809   _Vcvv 3109   <.cop 4038   dom cdm 5008   Rel wrel 5013   Fun wfun 5588   ` cfv 5594   1stc1st 6797   2ndc2nd 6798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-1st 6799  df-2nd 6800
This theorem is referenced by:  fcnvgreu  27662
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