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Theorem fnima 6632
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 → (𝐹𝐴) = ran 𝐹)

Proof of Theorem fnima
StepHypRef Expression
1 df-ima 5647 . 2 (𝐹𝐴) = ran (𝐹𝐴)
2 fnresdm 6621 . . 3 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
32rneqd 5894 . 2 (𝐹 Fn 𝐴 → ran (𝐹𝐴) = ran 𝐹)
41, 3eqtrid 2785 1 (𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  ran crn 5635  cres 5636  cima 5637   Fn wfn 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499  df-fn 6500
This theorem is referenced by:  infdifsn  9598  cardinfima  10038  alephfp  10049  dprdf1o  19816  dprd2db  19827  lmhmrnlss  20526  frlmlbs  21219  frlmup3  21222  ellspd  21224  mpfsubrg  21529  pf1subrg  21730  tgrest  22526  uniiccdif  24958  uniioombllem3  24965  dvgt0lem2  25383  f1rnen  31589  cycpmco2rn  32023  fedgmul  32383  zarclsint  32510  eulerpartlemn  33038  matunitlindflem2  36121  poimirlem15  36139  k0004lem1  42507  imasetpreimafvbijlemf  45679  fundcmpsurbijinjpreimafv  45685
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