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Theorem fndmin 5217
Description: Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
fndmin  F  Fn  G  Fn  dom  F  i^i  G  {  |  F `
 G `  }
Distinct variable groups:   , F   , G   ,

Proof of Theorem fndmin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dffn5im 5162 . . . . . 6  F  Fn  F  |->  F `
2 df-mpt 3811 . . . . . 6  |->  F `
 { <. ,  >.  |  F `  }
31, 2syl6eq 2085 . . . . 5  F  Fn  F  { <. , 
>.  |  F `  }
4 dffn5im 5162 . . . . . 6  G  Fn  G  |->  G `
5 df-mpt 3811 . . . . . 6  |->  G `
 { <. ,  >.  |  G `  }
64, 5syl6eq 2085 . . . . 5  G  Fn  G  { <. , 
>.  |  G `  }
73, 6ineqan12d 3134 . . . 4  F  Fn  G  Fn  F  i^i  G  { <. ,  >.  |  F `  }  i^i  { <. ,  >.  |  G `  }
8 inopab 4411 . . . 4  { <. ,  >.  |  F `  }  i^i  { <. ,  >.  |  G `  }  { <. ,  >.  |  F `  G `  }
97, 8syl6eq 2085 . . 3  F  Fn  G  Fn  F  i^i  G  { <. ,  >.  |  F `  G `  }
109dmeqd 4480 . 2  F  Fn  G  Fn  dom  F  i^i  G  dom  { <. ,  >.  |  F `  G `  }
11 anandi 524 . . . . . . . 8  F `  G `  F `  G `
1211exbii 1493 . . . . . . 7  F `  G `  F `  G `
13 19.42v 1783 . . . . . . 7  F `  G `  F `  G `
1412, 13bitr3i 175 . . . . . 6  F `  G `  F `  G `
15 funfvex 5135 . . . . . . . . 9  Fun  F  dom  F  F `  _V
16 eqeq1 2043 . . . . . . . . . 10  F `  G `  F `
 G `
1716ceqsexgv 2667 . . . . . . . . 9  F `  _V  F `  G `  F `  G `
1815, 17syl 14 . . . . . . . 8  Fun  F  dom  F  F `  G `  F `  G `
1918funfni 4942 . . . . . . 7  F  Fn  F `
 G `  F `
 G `
2019pm5.32da 425 . . . . . 6  F  Fn  F `  G `  F `  G `
2114, 20syl5bb 181 . . . . 5  F  Fn  F `  G `  F `  G `
2221abbidv 2152 . . . 4  F  Fn  {  |  F `  G `  }  {  |  F `  G `  }
23 dmopab 4489 . . . 4  dom  { <. ,  >.  |  F `  G `  }  {  |  F `  G `  }
24 df-rab 2309 . . . 4  {  |  F `
 G `  }  {  |  F `
 G `  }
2522, 23, 243eqtr4g 2094 . . 3  F  Fn  dom  {
<. ,  >.  |  F `  G `  }  {  |  F `
 G `  }
2625adantr 261 . 2  F  Fn  G  Fn  dom  { <. ,  >.  |  F `  G `  }  {  |  F `  G `  }
2710, 26eqtrd 2069 1  F  Fn  G  Fn  dom  F  i^i  G  {  |  F `
 G `  }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390   {cab 2023   {crab 2304   _Vcvv 2551    i^i cin 2910   {copab 3808    |-> cmpt 3809   dom cdm 4288   Fun wfun 4839    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-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-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  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-mpt 3811  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:  fneqeql  5218
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