MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fndmdifeq0 Unicode version

Theorem fndmdifeq0 5714
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 5712 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  {
x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) } )
21eqeq1d 2366 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( dom  ( F 
\  G )  =  (/) 
<->  { x  e.  A  |  ( F `  x )  =/=  ( G `  x ) }  =  (/) ) )
3 eqfnfv 5705 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
4 rabeq0 3552 . . . 4  |-  ( { x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) }  =  (/)  <->  A. x  e.  A  -.  ( F `  x
)  =/=  ( G `
 x ) )
5 nne 2525 . . . . 5  |-  ( -.  ( F `  x
)  =/=  ( G `
 x )  <->  ( F `  x )  =  ( G `  x ) )
65ralbii 2643 . . . 4  |-  ( A. x  e.  A  -.  ( F `  x )  =/=  ( G `  x )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
74, 6bitri 240 . . 3  |-  ( { x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) }  =  (/)  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
83, 7syl6rbbr 255 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( { x  e.  A  |  ( F `
 x )  =/=  ( G `  x
) }  =  (/)  <->  F  =  G ) )
92, 8bitrd 244 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 176    /\ wa 358    = wceq 1642    =/= wne 2521   A.wral 2619   {crab 2623    \ cdif 3225   (/)c0 3531   dom cdm 4771    Fn wfn 5332   ` cfv 5337
This theorem is referenced by:  wemapso  7356  wemapso2  7357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-fv 5345
  Copyright terms: Public domain W3C validator