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Theorem fndmdifeq0 5803
Description: The difference set of two functions is empty if and only if the functions are equal. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
fndmdifeq0  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( dom  ( F 
\  G )  =  (/) 
<->  F  =  G ) )

Proof of Theorem fndmdifeq0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fndmdif 5801 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  {
x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) } )
21eqeq1d 2420 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( dom  ( F 
\  G )  =  (/) 
<->  { x  e.  A  |  ( F `  x )  =/=  ( G `  x ) }  =  (/) ) )
3 eqfnfv 5794 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
4 rabeq0 3617 . . . 4  |-  ( { x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) }  =  (/)  <->  A. x  e.  A  -.  ( F `  x
)  =/=  ( G `
 x ) )
5 nne 2579 . . . . 5  |-  ( -.  ( F `  x
)  =/=  ( G `
 x )  <->  ( F `  x )  =  ( G `  x ) )
65ralbii 2698 . . . 4  |-  ( A. x  e.  A  -.  ( F `  x )  =/=  ( G `  x )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
74, 6bitri 241 . . 3  |-  ( { x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) }  =  (/)  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
83, 7syl6rbbr 256 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( { x  e.  A  |  ( F `
 x )  =/=  ( G `  x
) }  =  (/)  <->  F  =  G ) )
92, 8bitrd 245 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( dom  ( F 
\  G )  =  (/) 
<->  F  =  G ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    =/= wne 2575   A.wral 2674   {crab 2678    \ cdif 3285   (/)c0 3596   dom cdm 4845    Fn wfn 5416   ` cfv 5421
This theorem is referenced by:  wemapso  7484  wemapso2  7485
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-fv 5429
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