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Theorem fneqeql 5637
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 5626 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
2 eqcom 2189 . . . 4  |-  ( { x  e.  A  | 
( F `  x
)  =  ( G `
 x ) }  =  A  <->  A  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) } )
3 rabid2 2664 . . . 4  |-  ( A  =  { x  e.  A  |  ( F `
 x )  =  ( G `  x
) }  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
42, 3bitri 184 . . 3  |-  ( { x  e.  A  | 
( F `  x
)  =  ( G `
 x ) }  =  A  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
51, 4bitr4di 198 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  { x  e.  A  |  ( F `  x )  =  ( G `  x ) }  =  A ) )
6 fndmin 5636 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  i^i  G )  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) } )
76eqeq1d 2196 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( dom  ( F  i^i  G )  =  A  <->  { x  e.  A  |  ( F `  x )  =  ( G `  x ) }  =  A ) )
85, 7bitr4d 191 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 104    <-> wb 105    = wceq 1363   A.wral 2465   {crab 2469    i^i cin 3140   dom cdm 4638    Fn wfn 5223   ` cfv 5228
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236
This theorem is referenced by:  fneqeql2  5638  fnreseql  5639
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