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Theorem fnima 5564
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 4892 . 2  |-  ( F
" A )  =  ran  ( F  |`  A )
2 fnresdm 5555 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
32rneqd 5098 . 2  |-  ( F  Fn  A  ->  ran  ( F  |`  A )  =  ran  F )
41, 3syl5eq 2481 1  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   ran crn 4880    |` cres 4881   "cima 4882    Fn wfn 5450
This theorem is referenced by:  infdifsn  7612  carduniima  7978  cardinfima  7979  alephfp  7990  dprdf1o  15591  dprd2db  15602  lmhmrnlss  16127  tgrest  17224  uniiccdif  19471  uniioombllem3  19478  dvgt0lem2  19888  mpfsubrg  19962  pf1subrg  19969  frlmlbs  27227  frlmup3  27230  ellspd  27232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-rel 4886  df-cnv 4887  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-fun 5457  df-fn 5458
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