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Theorem fndmdifcom 5405
Description: The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
fndmdifcom  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  dom  ( G  \  F ) )

Proof of Theorem fndmdifcom
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 necom 2339 . . . 4  |-  ( ( F `  x )  =/=  ( G `  x )  <->  ( G `  x )  =/=  ( F `  x )
)
21a1i 9 . . 3  |-  ( x  e.  A  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  ( G `  x )  =/=  ( F `  x )
) )
32rabbiia 2604 . 2  |-  { x  e.  A  |  ( F `  x )  =/=  ( G `  x
) }  =  {
x  e.  A  | 
( G `  x
)  =/=  ( F `
 x ) }
4 fndmdif 5404 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  {
x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) } )
5 fndmdif 5404 . . 3  |-  ( ( G  Fn  A  /\  F  Fn  A )  ->  dom  ( G  \  F )  =  {
x  e.  A  | 
( G `  x
)  =/=  ( F `
 x ) } )
65ancoms 264 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( G  \  F )  =  {
x  e.  A  | 
( G `  x
)  =/=  ( F `
 x ) } )
73, 4, 63eqtr4a 2146 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  dom  ( G  \  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438    =/= wne 2255   {crab 2363    \ cdif 2996   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-in1 579  ax-in2 580  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-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  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-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: (None)
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