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Theorem fndmdif 5575
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 3234 . . . . 5  |-  ( F 
\  G )  C_  F
2 dmss 4788 . . . . 5  |-  ( ( F  \  G ) 
C_  F  ->  dom  ( F  \  G ) 
C_  dom  F )
31, 2ax-mp 5 . . . 4  |-  dom  ( F  \  G )  C_  dom  F
4 fndm 5272 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
54adantr 274 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  F  =  A )
63, 5sseqtrid 3178 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  dom  ( F  \  G )  C_  A
)
7 dfss1 3312 . . 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 2715 . . . . 5  |-  x  e. 
_V
109eldm 4786 . . . 4  |-  ( x  e.  dom  ( F 
\  G )  <->  E. y  x ( F  \  G ) y )
11 eqcom 2159 . . . . . . . 8  |-  ( ( F `  x )  =  ( G `  x )  <->  ( G `  x )  =  ( F `  x ) )
12 fnbrfvb 5512 . . . . . . . 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 468 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( F `  x
)  =  ( G `
 x )  <->  x G
( F `  x
) ) )
1514necon3abid 2366 . . . . 5  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( F `  x
)  =/=  ( G `
 x )  <->  -.  x G ( F `  x ) ) )
16 funfvex 5488 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
1716funfni 5273 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  e.  _V )
1817adantlr 469 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  ( F `  x )  e.  _V )
19 breq2 3971 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
x G y  <->  x G
( F `  x
) ) )
2019notbid 657 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  ( -.  x G y  <->  -.  x G ( F `  x ) ) )
2120ceqsexgv 2841 . . . . . 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 2159 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
24 fnbrfvb 5512 . . . . . . . . . 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 469 . . . . . . . 8  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  x F
y ) )
2726anbi1d 461 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  x  e.  A )  ->  (
( y  =  ( F `  x )  /\  -.  x G y )  <->  ( x F y  /\  -.  x G y ) ) )
28 brdif 4020 . . . . . . 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 1805 . . . . 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 3316 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A  i^i  dom  ( F  \  G ) )  =  { x  e.  A  |  ( F `  x )  =/=  ( G `  x
) } )
348, 33eqtr3d 2192 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 1335   E.wex 1472    e. wcel 2128    =/= wne 2327   {crab 2439   _Vcvv 2712    \ cdif 3099    i^i cin 3101    C_ wss 3102   class class class wbr 3967   dom cdm 4589    Fn wfn 5168   ` cfv 5173
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4029  df-id 4256  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-iota 5138  df-fun 5175  df-fn 5176  df-fv 5181
This theorem is referenced by:  fndmdifcom  5576
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