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Theorem fnima 5705
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 5018 . 2  |-  ( F
" A )  =  ran  ( F  |`  A )
2 fnresdm 5696 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
32rneqd 5236 . 2  |-  ( F  Fn  A  ->  ran  ( F  |`  A )  =  ran  F )
41, 3syl5eq 2520 1  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   ran crn 5006    |` cres 5007   "cima 5008    Fn wfn 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-cnv 5013  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-fun 5596  df-fn 5597
This theorem is referenced by:  infdifsn  8085  carduniima  8489  cardinfima  8490  alephfp  8501  dprdf1o  16951  dprd2db  16964  lmhmrnlss  17567  mpfsubrg  18071  pf1subrg  18254  frlmlbs  18700  frlmup3  18703  ellspd  18705  ellspdOLD  18706  tgrest  19528  uniiccdif  21855  uniioombllem3  21862  dvgt0lem2  22272  eulerpartlemn  28145
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