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Theorem fnima 5362
Description: The image of a function's domain is its range. (Contributed by NM, 4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fnima  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )

Proof of Theorem fnima
StepHypRef Expression
1 df-ima 4702 . 2  |-  ( F
" A )  =  ran  ( F  |`  A )
2 fnresdm 5353 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
32rneqd 4906 . 2  |-  ( F  Fn  A  ->  ran  ( F  |`  A )  =  ran  F )
41, 3syl5eq 2327 1  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250
This theorem is referenced by:  infdifsn  7357  carduniima  7723  cardinfima  7724  alephfp  7735  dprdf1o  15267  dprd2db  15278  lmhmrnlss  15807  tgrest  16890  uniiccdif  18933  uniioombllem3  18940  dvgt0lem2  19350  mpfsubrg  19424  pf1subrg  19431  caytr  25400  frlmlbs  27249  frlmup3  27252  ellspd  27254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258
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