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Theorem fneqeql 5407
Description: Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Assertion
Ref Expression
fneqeql  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  dom  ( F  i^i  G
)  =  A ) )

Proof of Theorem fneqeql
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqfnfv 5397 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
2 eqcom 2090 . . . 4  |-  ( { x  e.  A  | 
( F `  x
)  =  ( G `
 x ) }  =  A  <->  A  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) } )
3 rabid2 2543 . . . 4  |-  ( A  =  { x  e.  A  |  ( F `
 x )  =  ( G `  x
) }  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
42, 3bitri 182 . . 3  |-  ( { x  e.  A  | 
( F `  x
)  =  ( G `
 x ) }  =  A  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
51, 4syl6bbr 196 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  { x  e.  A  |  ( F `  x )  =  ( G `  x ) }  =  A ) )
6 fndmin 5406 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  i^i  G )  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) } )
76eqeq1d 2096 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( dom  ( F  i^i  G )  =  A  <->  { x  e.  A  |  ( F `  x )  =  ( G `  x ) }  =  A ) )
85, 7bitr4d 189 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  dom  ( F  i^i  G
)  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   A.wral 2359   {crab 2363    i^i cin 2998   dom cdm 4438    Fn wfn 5010   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023
This theorem is referenced by:  fneqeql2  5408  fnreseql  5409
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