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Theorem fneqeql 5633
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 5622 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
2 eqcom 2285 . . . 4  |-  ( { x  e.  A  | 
( F `  x
)  =  ( G `
 x ) }  =  A  <->  A  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) } )
3 rabid2 2717 . . . 4  |-  ( A  =  { x  e.  A  |  ( F `
 x )  =  ( G `  x
) }  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
42, 3bitri 240 . . 3  |-  ( { x  e.  A  | 
( F `  x
)  =  ( G `
 x ) }  =  A  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
51, 4syl6bbr 254 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  { x  e.  A  |  ( F `  x )  =  ( G `  x ) }  =  A ) )
6 fndmin 5632 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  i^i  G )  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) } )
76eqeq1d 2291 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( dom  ( F  i^i  G )  =  A  <->  { x  e.  A  |  ( F `  x )  =  ( G `  x ) }  =  A ) )
85, 7bitr4d 247 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    <-> wb 176    /\ wa 358    = wceq 1623   A.wral 2543   {crab 2547    i^i cin 3151   dom cdm 4689    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  fneqeql2  5634  fnreseql  5635  lspextmo  15813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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