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Theorem fnima 6472
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 5562 . 2 (𝐹𝐴) = ran (𝐹𝐴)
2 fnresdm 6460 . . 3 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
32rneqd 5802 . 2 (𝐹 Fn 𝐴 → ran (𝐹𝐴) = ran 𝐹)
41, 3syl5eq 2868 1 (𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  ran crn 5550  cres 5551  cima 5552   Fn wfn 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-fun 6351  df-fn 6352
This theorem is referenced by:  infdifsn  9109  cardinfima  9512  alephfp  9523  dprdf1o  19085  dprd2db  19096  lmhmrnlss  19753  mpfsubrg  20246  pf1subrg  20441  frlmlbs  20871  frlmup3  20874  ellspd  20876  tgrest  21697  uniiccdif  24108  uniioombllem3  24115  dvgt0lem2  24529  f1rnen  30303  cycpmco2rn  30695  fedgmul  30927  eulerpartlemn  31539  matunitlindflem2  34771  poimirlem15  34789  k0004lem1  40377
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