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Theorem fndmdifcom 5646
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 2540 . . . 4  |-  ( ( F `  x )  =/=  ( G `  x )  <->  ( G `  x )  =/=  ( F `  x )
)
21a1i 10 . . 3  |-  ( x  e.  A  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  ( G `  x )  =/=  ( F `  x )
) )
32rabbiia 2791 . 2  |-  { x  e.  A  |  ( F `  x )  =/=  ( G `  x
) }  =  {
x  e.  A  | 
( G `  x
)  =/=  ( F `
 x ) }
4 fndmdif 5645 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  {
x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) } )
5 fndmdif 5645 . . 3  |-  ( ( G  Fn  A  /\  F  Fn  A )  ->  dom  ( G  \  F )  =  {
x  e.  A  | 
( G `  x
)  =/=  ( F `
 x ) } )
65ancoms 439 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( G  \  F )  =  {
x  e.  A  | 
( G `  x
)  =/=  ( F `
 x ) } )
73, 4, 63eqtr4a 2354 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  dom  ( G  \  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560    \ cdif 3162   dom cdm 4705    Fn wfn 5266   ` cfv 5271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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