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Theorem fndmdif 5967
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 3617 . . . . 5  |-  ( F 
\  G )  C_  F
2 dmss 5191 . . . . 5  |-  ( ( F  \  G ) 
C_  F  ->  dom  ( F  \  G ) 
C_  dom  F )
31, 2ax-mp 5 . . . 4  |-  dom  ( F  \  G )  C_  dom  F
4 fndm 5662 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
54adantr 463 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  F  =  A )
63, 5syl5sseq 3537 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  C_  A
)
7 dfss1 3689 . . 3  |-  ( dom  ( F  \  G
)  C_  A  <->  ( A  i^i  dom  ( F  \  G ) )  =  dom  ( F  \  G ) )
86, 7sylib 196 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A  i^i  dom  ( F  \  G ) )  =  dom  ( F  \  G ) )
9 vex 3109 . . . . 5  |-  x  e. 
_V
109eldm 5189 . . . 4  |-  ( x  e.  dom  ( F 
\  G )  <->  E. y  x ( F  \  G ) y )
11 eqcom 2463 . . . . . . . . 9  |-  ( ( F `  x )  =  ( G `  x )  <->  ( G `  x )  =  ( F `  x ) )
12 fnbrfvb 5888 . . . . . . . . 9  |-  ( ( G  Fn  A  /\  x  e.  A )  ->  ( ( G `  x )  =  ( F `  x )  <-> 
x G ( F `
 x ) ) )
1311, 12syl5bb 257 . . . . . . . 8  |-  ( ( G  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  ( G `  x )  <-> 
x G ( F `
 x ) ) )
1413adantll 711 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( F `  x
)  =  ( G `
 x )  <->  x G
( F `  x
) ) )
1514necon3abid 2700 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  -.  x G ( F `  x ) ) )
16 fvex 5858 . . . . . . 7  |-  ( F `
 x )  e. 
_V
17 breq2 4443 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
x G y  <->  x G
( F `  x
) ) )
1817notbid 292 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  ( -.  x G y  <->  -.  x G ( F `  x ) ) )
1916, 18ceqsexv 3143 . . . . . 6  |-  ( E. y ( y  =  ( F `  x
)  /\  -.  x G y )  <->  -.  x G ( F `  x ) )
2015, 19syl6bbr 263 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  E. y
( y  =  ( F `  x )  /\  -.  x G y ) ) )
21 eqcom 2463 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
22 fnbrfvb 5888 . . . . . . . . . 10  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
2321, 22syl5bb 257 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( y  =  ( F `  x )  <-> 
x F y ) )
2423adantlr 712 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  x F
y ) )
2524anbi1d 702 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( y  =  ( F `  x )  /\  -.  x G y )  <->  ( x F y  /\  -.  x G y ) ) )
26 brdif 4489 . . . . . . 7  |-  ( x ( F  \  G
) y  <->  ( x F y  /\  -.  x G y ) )
2725, 26syl6bbr 263 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( y  =  ( F `  x )  /\  -.  x G y )  <->  x ( F  \  G ) y ) )
2827exbidv 1719 . . . . 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 254 . . . 4  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  ( E. y  x ( F  \  G ) y  <-> 
( F `  x
)  =/=  ( G `
 x ) ) )
3010, 29syl5bb 257 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
x  e.  dom  ( F  \  G )  <->  ( F `  x )  =/=  ( G `  x )
) )
3130rabbi2dva 3692 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A  i^i  dom  ( F  \  G ) )  =  { x  e.  A  |  ( F `  x )  =/=  ( G `  x
) } )
328, 31eqtr3d 2497 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 184    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   {crab 2808    \ cdif 3458    i^i cin 3460    C_ wss 3461   class class class wbr 4439   dom cdm 4988    Fn wfn 5565   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578
This theorem is referenced by:  fndmdifcom  5968  fndmdifeq0  5969  fndifnfp  6076  wemapsolem  7967  wemapso2OLD  7969  wemapso2lem  7970  dsmmbas2  18941  frlmbas  18959  frlmbasOLD  18960  ptcmplem2  20719
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