ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fneqeql Unicode version

Theorem fneqeql 5303
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 5293 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
2 eqcom 2058 . . . 4  |-  ( { x  e.  A  | 
( F `  x
)  =  ( G `
 x ) }  =  A  <->  A  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) } )
3 rabid2 2503 . . . 4  |-  ( A  =  { x  e.  A  |  ( F `
 x )  =  ( G `  x
) }  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
42, 3bitri 177 . . 3  |-  ( { x  e.  A  | 
( F `  x
)  =  ( G `
 x ) }  =  A  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
51, 4syl6bbr 191 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  { x  e.  A  |  ( F `  x )  =  ( G `  x ) }  =  A ) )
6 fndmin 5302 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  i^i  G )  =  { x  e.  A  |  ( F `  x )  =  ( G `  x ) } )
76eqeq1d 2064 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( dom  ( F  i^i  G )  =  A  <->  { x  e.  A  |  ( F `  x )  =  ( G `  x ) }  =  A ) )
85, 7bitr4d 184 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 101    <-> wb 102    = wceq 1259   A.wral 2323   {crab 2327    i^i cin 2944   dom cdm 4373    Fn wfn 4925   ` cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938
This theorem is referenced by:  fneqeql2  5304  fnreseql  5305
  Copyright terms: Public domain W3C validator