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Theorem fndmdif 5215
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  G  Fn  dom  F  \  G  {  |  F `  =/=  G `  }
Distinct variable groups:   , F   , G   ,

Proof of Theorem fndmdif
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 difss 3064 . . . . 5  F 
\  G  C_  F
2 dmss 4477 . . . . 5  F  \  G 
C_  F  dom  F  \  G 
C_  dom  F
31, 2ax-mp 7 . . . 4  dom  F  \  G  C_  dom  F
4 fndm 4941 . . . . 5  F  Fn  dom  F
54adantr 261 . . . 4  F  Fn  G  Fn  dom  F
63, 5syl5sseq 2987 . . 3  F  Fn  G  Fn  dom  F  \  G  C_
7 dfss1 3135 . . 3  dom  F  \  G  C_  i^i  dom  F  \  G 
dom  F  \  G
86, 7sylib 127 . 2  F  Fn  G  Fn  i^i  dom  F  \  G  dom  F  \  G
9 vex 2554 . . . . 5 
_V
109eldm 4475 . . . 4  dom  F 
\  G  F  \  G
11 eqcom 2039 . . . . . . . 8  F `  G `  G `  F `
12 fnbrfvb 5157 . . . . . . . 8  G  Fn  G `  F `  G F `
1311, 12syl5bb 181 . . . . . . 7  G  Fn  F `  G `  G F `
1413adantll 445 . . . . . 6  F  Fn  G  Fn  F `
 G `  G F `
1514necon3abid 2238 . . . . 5  F  Fn  G  Fn  F `
 =/=  G `  G F `
16 funfvex 5135 . . . . . . . 8  Fun  F  dom  F  F `  _V
1716funfni 4942 . . . . . . 7  F  Fn  F `  _V
1817adantlr 446 . . . . . 6  F  Fn  G  Fn  F `  _V
19 breq2 3759 . . . . . . . 8  F `  G  G F `
2019notbid 591 . . . . . . 7  F `  G  G F `
2120ceqsexgv 2667 . . . . . 6  F `  _V  F `  G  G F `
2218, 21syl 14 . . . . 5  F  Fn  G  Fn  F `  G  G F `
23 eqcom 2039 . . . . . . . . . 10  F `  F `
24 fnbrfvb 5157 . . . . . . . . . 10  F  Fn  F `  F
2523, 24syl5bb 181 . . . . . . . . 9  F  Fn  F `  F
2625adantlr 446 . . . . . . . 8  F  Fn  G  Fn  F `  F
2726anbi1d 438 . . . . . . 7  F  Fn  G  Fn  F `  G  F  G
28 brdif 3803 . . . . . . 7  F  \  G  F  G
2927, 28syl6bbr 187 . . . . . 6  F  Fn  G  Fn  F `  G  F  \  G
3029exbidv 1703 . . . . 5  F  Fn  G  Fn  F `  G  F  \  G
3115, 22, 303bitr2rd 206 . . . 4  F  Fn  G  Fn  F  \  G  F `  =/=  G `
3210, 31syl5bb 181 . . 3  F  Fn  G  Fn 
dom  F  \  G  F `  =/=  G `
3332rabbi2dva 3139 . 2  F  Fn  G  Fn  i^i  dom  F  \  G  {  |  F `
 =/=  G `  }
348, 33eqtr3d 2071 1  F  Fn  G  Fn  dom  F  \  G  {  |  F `  =/=  G `  }
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390    =/= wne 2201   {crab 2304   _Vcvv 2551    \ cdif 2908    i^i cin 2910    C_ wss 2911   class class class wbr 3755   dom cdm 4288    Fn wfn 4840   ` cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  fndmdifcom  5216
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