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Theorem fndmdifcom 5617
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 2431 . . . 4  |-  ( ( F `  x )  =/=  ( G `  x )  <->  ( G `  x )  =/=  ( F `  x )
)
21a1i 9 . . 3  |-  ( x  e.  A  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  ( G `  x )  =/=  ( F `  x )
) )
32rabbiia 2722 . 2  |-  { x  e.  A  |  ( F `  x )  =/=  ( G `  x
) }  =  {
x  e.  A  | 
( G `  x
)  =/=  ( F `
 x ) }
4 fndmdif 5616 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  {
x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) } )
5 fndmdif 5616 . . 3  |-  ( ( G  Fn  A  /\  F  Fn  A )  ->  dom  ( G  \  F )  =  {
x  e.  A  | 
( G `  x
)  =/=  ( F `
 x ) } )
65ancoms 268 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( G  \  F )  =  {
x  e.  A  | 
( G `  x
)  =/=  ( F `
 x ) } )
73, 4, 63eqtr4a 2236 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  dom  ( G  \  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148    =/= wne 2347   {crab 2459    \ cdif 3126   dom cdm 4622    Fn wfn 5206   ` cfv 5211
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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219
This theorem is referenced by: (None)
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