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Theorem fvelimab 5172
Description: Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)
Assertion
Ref Expression
fvelimab  F  Fn  C_  C  F "  F `  C
Distinct variable groups:   ,   , C   , F
Allowed substitution hint:   ()

Proof of Theorem fvelimab
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2560 . . . 4  C  F "  C 
_V
21anim2i 324 . . 3  F  Fn  C_  C  F "  F  Fn  C_  C  _V
3 ssel2 2934 . . . . . . . 8  C_
4 funfvex 5135 . . . . . . . . 9  Fun  F  dom  F  F `  _V
54funfni 4942 . . . . . . . 8  F  Fn  F `  _V
63, 5sylan2 270 . . . . . . 7  F  Fn  C_  F `  _V
76anassrs 380 . . . . . 6  F  Fn  C_  F `  _V
8 eleq1 2097 . . . . . 6  F `  C  F `  _V  C 
_V
97, 8syl5ibcom 144 . . . . 5  F  Fn  C_  F `
 C  C 
_V
109rexlimdva 2427 . . . 4  F  Fn  C_  F `  C  C  _V
1110imdistani 419 . . 3  F  Fn  C_  F `  C  F  Fn  C_  C  _V
12 eleq1 2097 . . . . . . 7  C  F "  C  F "
13 eqeq2 2046 . . . . . . . 8  C  F `  F `
 C
1413rexbidv 2321 . . . . . . 7  C  F `  F `  C
1512, 14bibi12d 224 . . . . . 6  C  F "  F `  C  F "  F `  C
1615imbi2d 219 . . . . 5  C  F  Fn  C_  F "  F `  F  Fn  C_  C  F "  F `  C
17 fnfun 4939 . . . . . . . 8  F  Fn  Fun  F
1817adantr 261 . . . . . . 7  F  Fn  C_ 
Fun  F
19 fndm 4941 . . . . . . . . 9  F  Fn  dom  F
2019sseq2d 2967 . . . . . . . 8  F  Fn  C_  dom  F  C_
2120biimpar 281 . . . . . . 7  F  Fn  C_  C_  dom  F
22 dfimafn 5165 . . . . . . 7  Fun  F  C_ 
dom  F  F "  {  |  F `  }
2318, 21, 22syl2anc 391 . . . . . 6  F  Fn  C_  F "  {  |  F `  }
2423abeq2d 2147 . . . . 5  F  Fn  C_  F "  F `
2516, 24vtoclg 2607 . . . 4  C  _V  F  Fn  C_  C  F "  F `  C
2625impcom 116 . . 3  F  Fn  C_  C  _V  C  F "  F `  C
272, 11, 26pm5.21nd 824 . 2  F  Fn  C_  C  F "  F `  C
28 fveq2 5121 . . . 4  F `  F `
2928eqeq1d 2045 . . 3  F `  C  F `
 C
3029cbvrexv 2528 . 2  F `  C  F `  C
3127, 30syl6bb 185 1  F  Fn  C_  C  F "  F `  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390   {cab 2023  wrex 2301   _Vcvv 2551    C_ wss 2911   dom cdm 4288   "cima 4291   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-bnd 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-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-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  ssimaex  5177  rexima  5337  ralima  5338  f1elima  5355  ovelimab  5593
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