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Theorem fndmdifeq0 5978
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 5976 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  {
x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) } )
21eqeq1d 2445 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( dom  ( F 
\  G )  =  (/) 
<->  { x  e.  A  |  ( F `  x )  =/=  ( G `  x ) }  =  (/) ) )
3 eqfnfv 5966 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
4 rabeq0 3793 . . . 4  |-  ( { x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) }  =  (/)  <->  A. x  e.  A  -.  ( F `  x
)  =/=  ( G `
 x ) )
5 nne 2644 . . . . 5  |-  ( -.  ( F `  x
)  =/=  ( G `
 x )  <->  ( F `  x )  =  ( G `  x ) )
65ralbii 2874 . . . 4  |-  ( A. x  e.  A  -.  ( F `  x )  =/=  ( G `  x )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
74, 6bitri 249 . . 3  |-  ( { x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) }  =  (/)  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
83, 7syl6rbbr 264 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( { x  e.  A  |  ( F `
 x )  =/=  ( G `  x
) }  =  (/)  <->  F  =  G ) )
92, 8bitrd 253 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( dom  ( F 
\  G )  =  (/) 
<->  F  =  G ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    =/= wne 2638   A.wral 2793   {crab 2797    \ cdif 3458   (/)c0 3770   dom cdm 4989    Fn wfn 5573   ` cfv 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-fv 5586
This theorem is referenced by:  wemapso  7979  wemapso2OLD  7980  wemapso2lem  7981
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