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Theorem fnima 5526
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 4849 . 2  |-  ( F
" A )  =  ran  ( F  |`  A )
2 fnresdm 5517 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
32rneqd 5063 . 2  |-  ( F  Fn  A  ->  ran  ( F  |`  A )  =  ran  F )
41, 3syl5eq 2485 1  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364   ran crn 4837    |` cres 4838   "cima 4839    Fn wfn 5410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-xp 4842  df-rel 4843  df-cnv 4844  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-fun 5417  df-fn 5418
This theorem is referenced by:  infdifsn  7858  carduniima  8262  cardinfima  8263  alephfp  8274  dprdf1o  16519  dprd2db  16532  lmhmrnlss  17109  mpfsubrg  17594  pf1subrg  17751  frlmlbs  18184  frlmup3  18187  ellspd  18189  ellspdOLD  18190  tgrest  18722  uniiccdif  21017  uniioombllem3  21024  dvgt0lem2  21434  eulerpartlemn  26694
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