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Theorem fndmdif 5601
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 3253 . . . . 5  |-  ( F 
\  G )  C_  F
2 dmss 4810 . . . . 5  |-  ( ( F  \  G ) 
C_  F  ->  dom  ( F  \  G ) 
C_  dom  F )
31, 2ax-mp 5 . . . 4  |-  dom  ( F  \  G )  C_  dom  F
4 fndm 5297 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
54adantr 274 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  F  =  A )
63, 5sseqtrid 3197 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  C_  A
)
7 dfss1 3331 . . 3  |-  ( dom  ( F  \  G
)  C_  A  <->  ( A  i^i  dom  ( F  \  G ) )  =  dom  ( F  \  G ) )
86, 7sylib 121 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A  i^i  dom  ( F  \  G ) )  =  dom  ( F  \  G ) )
9 vex 2733 . . . . 5  |-  x  e. 
_V
109eldm 4808 . . . 4  |-  ( x  e.  dom  ( F 
\  G )  <->  E. y  x ( F  \  G ) y )
11 eqcom 2172 . . . . . . . 8  |-  ( ( F `  x )  =  ( G `  x )  <->  ( G `  x )  =  ( F `  x ) )
12 fnbrfvb 5537 . . . . . . . 8  |-  ( ( G  Fn  A  /\  x  e.  A )  ->  ( ( G `  x )  =  ( F `  x )  <-> 
x G ( F `
 x ) ) )
1311, 12syl5bb 191 . . . . . . 7  |-  ( ( G  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  ( G `  x )  <-> 
x G ( F `
 x ) ) )
1413adantll 473 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( F `  x
)  =  ( G `
 x )  <->  x G
( F `  x
) ) )
1514necon3abid 2379 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  -.  x G ( F `  x ) ) )
16 funfvex 5513 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
1716funfni 5298 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  e.  _V )
1817adantlr 474 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  ( F `  x )  e.  _V )
19 breq2 3993 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
x G y  <->  x G
( F `  x
) ) )
2019notbid 662 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  ( -.  x G y  <->  -.  x G ( F `  x ) ) )
2120ceqsexgv 2859 . . . . . 6  |-  ( ( F `  x )  e.  _V  ->  ( E. y ( y  =  ( F `  x
)  /\  -.  x G y )  <->  -.  x G ( F `  x ) ) )
2218, 21syl 14 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  ( E. y ( y  =  ( F `  x
)  /\  -.  x G y )  <->  -.  x G ( F `  x ) ) )
23 eqcom 2172 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
24 fnbrfvb 5537 . . . . . . . . . 10  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
2523, 24syl5bb 191 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( y  =  ( F `  x )  <-> 
x F y ) )
2625adantlr 474 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  x F
y ) )
2726anbi1d 462 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( y  =  ( F `  x )  /\  -.  x G y )  <->  ( x F y  /\  -.  x G y ) ) )
28 brdif 4042 . . . . . . 7  |-  ( x ( F  \  G
) y  <->  ( x F y  /\  -.  x G y ) )
2927, 28bitr4di 197 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( y  =  ( F `  x )  /\  -.  x G y )  <->  x ( F  \  G ) y ) )
3029exbidv 1818 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  ( E. y ( y  =  ( F `  x
)  /\  -.  x G y )  <->  E. y  x ( F  \  G ) y ) )
3115, 22, 303bitr2rd 216 . . . 4  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  ( E. y  x ( F  \  G ) y  <-> 
( F `  x
)  =/=  ( G `
 x ) ) )
3210, 31syl5bb 191 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
x  e.  dom  ( F  \  G )  <->  ( F `  x )  =/=  ( G `  x )
) )
3332rabbi2dva 3335 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A  i^i  dom  ( F  \  G ) )  =  { x  e.  A  |  ( F `  x )  =/=  ( G `  x
) } )
348, 33eqtr3d 2205 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  =  {
x  e.  A  | 
( F `  x
)  =/=  ( G `
 x ) } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348   E.wex 1485    e. wcel 2141    =/= wne 2340   {crab 2452   _Vcvv 2730    \ cdif 3118    i^i cin 3120    C_ wss 3121   class class class wbr 3989   dom cdm 4611    Fn wfn 5193   ` cfv 5198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206
This theorem is referenced by:  fndmdifcom  5602
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