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Theorem fndmdif 5945
Description: Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
fndmdif  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  {
x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) } )
Distinct variable groups:    x, F    x, G    x, A

Proof of Theorem fndmdif
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 difss 3535 . . . . 5  |-  ( F 
\  G )  C_  F
2 dmss 4996 . . . . 5  |-  ( ( F  \  G ) 
C_  F  ->  dom  ( F  \  G ) 
C_  dom  F )
31, 2ax-mp 5 . . . 4  |-  dom  ( F  \  G )  C_  dom  F
4 fndm 5636 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
54adantr 466 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  F  =  A )
63, 5syl5sseq 3455 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  C_  A
)
7 dfss1 3610 . . 3  |-  ( dom  ( F  \  G
)  C_  A  <->  ( A  i^i  dom  ( F  \  G ) )  =  dom  ( F  \  G ) )
86, 7sylib 199 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A  i^i  dom  ( F  \  G ) )  =  dom  ( F  \  G ) )
9 vex 3025 . . . . 5  |-  x  e. 
_V
109eldm 4994 . . . 4  |-  ( x  e.  dom  ( F 
\  G )  <->  E. y  x ( F  \  G ) y )
11 eqcom 2435 . . . . . . . . 9  |-  ( ( F `  x )  =  ( G `  x )  <->  ( G `  x )  =  ( F `  x ) )
12 fnbrfvb 5865 . . . . . . . . 9  |-  ( ( G  Fn  A  /\  x  e.  A )  ->  ( ( G `  x )  =  ( F `  x )  <-> 
x G ( F `
 x ) ) )
1311, 12syl5bb 260 . . . . . . . 8  |-  ( ( G  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  ( G `  x )  <-> 
x G ( F `
 x ) ) )
1413adantll 718 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( F `  x
)  =  ( G `
 x )  <->  x G
( F `  x
) ) )
1514necon3abid 2637 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  -.  x G ( F `  x ) ) )
16 fvex 5835 . . . . . . 7  |-  ( F `
 x )  e. 
_V
17 breq2 4370 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
x G y  <->  x G
( F `  x
) ) )
1817notbid 295 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  ( -.  x G y  <->  -.  x G ( F `  x ) ) )
1916, 18ceqsexv 3060 . . . . . 6  |-  ( E. y ( y  =  ( F `  x
)  /\  -.  x G y )  <->  -.  x G ( F `  x ) )
2015, 19syl6bbr 266 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  E. y
( y  =  ( F `  x )  /\  -.  x G y ) ) )
21 eqcom 2435 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
22 fnbrfvb 5865 . . . . . . . . . 10  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
2321, 22syl5bb 260 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( y  =  ( F `  x )  <-> 
x F y ) )
2423adantlr 719 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  x F
y ) )
2524anbi1d 709 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( y  =  ( F `  x )  /\  -.  x G y )  <->  ( x F y  /\  -.  x G y ) ) )
26 brdif 4417 . . . . . . 7  |-  ( x ( F  \  G
) y  <->  ( x F y  /\  -.  x G y ) )
2725, 26syl6bbr 266 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( y  =  ( F `  x )  /\  -.  x G y )  <->  x ( F  \  G ) y ) )
2827exbidv 1762 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  ( E. y ( y  =  ( F `  x
)  /\  -.  x G y )  <->  E. y  x ( F  \  G ) y ) )
2920, 28bitr2d 257 . . . 4  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  ( E. y  x ( F  \  G ) y  <-> 
( F `  x
)  =/=  ( G `
 x ) ) )
3010, 29syl5bb 260 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
x  e.  dom  ( F  \  G )  <->  ( F `  x )  =/=  ( G `  x )
) )
3130rabbi2dva 3613 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A  i^i  dom  ( F  \  G ) )  =  { x  e.  A  |  ( F `  x )  =/=  ( G `  x
) } )
328, 31eqtr3d 2464 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  {
x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872    =/= wne 2599   {crab 2718    \ cdif 3376    i^i cin 3378    C_ wss 3379   class class class wbr 4366   dom cdm 4796    Fn wfn 5539   ` cfv 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-iota 5508  df-fun 5546  df-fn 5547  df-fv 5552
This theorem is referenced by:  fndmdifcom  5946  fndmdifeq0  5947  fndifnfp  6052  wemapsolem  8018  wemapso2lem  8020  dsmmbas2  19242  frlmbas  19260  ptcmplem2  21010
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