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Theorem fndmdifcom 5827
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 2679 . . . 4  |-  ( ( F `  x )  =/=  ( G `  x )  <->  ( G `  x )  =/=  ( F `  x )
)
21a1i 11 . . 3  |-  ( x  e.  A  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  ( G `  x )  =/=  ( F `  x )
) )
32rabbiia 2938 . 2  |-  { x  e.  A  |  ( F `  x )  =/=  ( G `  x
) }  =  {
x  e.  A  | 
( G `  x
)  =/=  ( F `
 x ) }
4 fndmdif 5826 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  {
x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) } )
5 fndmdif 5826 . . 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 2493 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 1652    e. wcel 1725    =/= wne 2598   {crab 2701    \ cdif 3309   dom cdm 4870    Fn wfn 5441   ` cfv 5446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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