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Theorem fvelrnb 5164
Description: A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.)
Assertion
Ref Expression
fvelrnb  F  Fn  ran  F  F `
Distinct variable groups:   ,   ,   , F

Proof of Theorem fvelrnb
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-rex 2306 . . . 4  F `  F `
2 19.41v 1779 . . . . 5  F `
 F  Fn  F `  F  Fn
3 simpl 102 . . . . . . . . . 10  F `
43anim1i 323 . . . . . . . . 9  F `  F  Fn  F  Fn
54ancomd 254 . . . . . . . 8  F `  F  Fn  F  Fn
6 funfvex 5135 . . . . . . . . 9  Fun  F  dom  F  F `  _V
76funfni 4942 . . . . . . . 8  F  Fn  F `  _V
85, 7syl 14 . . . . . . 7  F `  F  Fn  F `  _V
9 simpr 103 . . . . . . . . 9  F `  F `
109eleq1d 2103 . . . . . . . 8  F `  F `  _V  _V
1110adantr 261 . . . . . . 7  F `  F  Fn  F `

_V  _V
128, 11mpbid 135 . . . . . 6  F `  F  Fn  _V
1312exlimiv 1486 . . . . 5  F `
 F  Fn  _V
142, 13sylbir 125 . . . 4  F `
 F  Fn  _V
151, 14sylanb 268 . . 3  F `  F  Fn  _V
1615expcom 109 . 2  F  Fn  F `  _V
17 fnrnfv 5163 . . . 4  F  Fn  ran  F  {  |  F `  }
1817eleq2d 2104 . . 3  F  Fn  ran  F  {  |  F `  }
19 eqeq1 2043 . . . . . 6  F `  F `
20 eqcom 2039 . . . . . 6  F `  F `
2119, 20syl6bb 185 . . . . 5  F `  F `
2221rexbidv 2321 . . . 4  F `  F `
2322elab3g 2687 . . 3  F `  _V  {  |  F `  }  F `
2418, 23sylan9bbr 436 . 2  F `  _V  F  Fn  ran  F  F `
2516, 24mpancom 399 1  F  Fn  ran  F  F `
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390   {cab 2023  wrex 2301   _Vcvv 2551   ran crn 4289    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-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  chfnrn  5221  rexrn  5247  ralrn  5248  elrnrexdmb  5250  ffnfv  5266  fconstfvm  5322  elunirn  5348  isoini  5400  reldm  5754  uzn0  8244  frec2uzrand  8852  frecuzrdgfn  8859
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