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Theorem fndmdifcom 5775
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 2632 . . . 4  |-  ( ( F `  x )  =/=  ( G `  x )  <->  ( G `  x )  =/=  ( F `  x )
)
21a1i 11 . . 3  |-  ( x  e.  A  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  ( G `  x )  =/=  ( F `  x )
) )
32rabbiia 2890 . 2  |-  { x  e.  A  |  ( F `  x )  =/=  ( G `  x
) }  =  {
x  e.  A  | 
( G `  x
)  =/=  ( F `
 x ) }
4 fndmdif 5774 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  {
x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) } )
5 fndmdif 5774 . . 3  |-  ( ( G  Fn  A  /\  F  Fn  A )  ->  dom  ( G  \  F )  =  {
x  e.  A  | 
( G `  x
)  =/=  ( F `
 x ) } )
65ancoms 440 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( G  \  F )  =  {
x  e.  A  | 
( G `  x
)  =/=  ( F `
 x ) } )
73, 4, 63eqtr4a 2446 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  dom  ( G  \  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   {crab 2654    \ cdif 3261   dom cdm 4819    Fn wfn 5390   ` cfv 5395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-iota 5359  df-fun 5397  df-fn 5398  df-fv 5403
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