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Theorem fnima 5534
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 4858 . 2  |-  ( F
" A )  =  ran  ( F  |`  A )
2 fnresdm 5525 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
32rneqd 5072 . 2  |-  ( F  Fn  A  ->  ran  ( F  |`  A )  =  ran  F )
41, 3syl5eq 2487 1  |-  ( F  Fn  A  ->  ( F " A )  =  ran  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   ran crn 4846    |` cres 4847   "cima 4848    Fn wfn 5418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-fun 5425  df-fn 5426
This theorem is referenced by:  infdifsn  7867  carduniima  8271  cardinfima  8272  alephfp  8283  dprdf1o  16534  dprd2db  16547  lmhmrnlss  17136  mpfsubrg  17623  pf1subrg  17787  frlmlbs  18230  frlmup3  18233  ellspd  18235  ellspdOLD  18236  tgrest  18768  uniiccdif  21063  uniioombllem3  21070  dvgt0lem2  21480  eulerpartlemn  26769
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