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Theorem fnima 6171
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 5279 . 2 (𝐹𝐴) = ran (𝐹𝐴)
2 fnresdm 6161 . . 3 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
32rneqd 5508 . 2 (𝐹 Fn 𝐴 → ran (𝐹𝐴) = ran 𝐹)
41, 3syl5eq 2806 1 (𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  ran crn 5267  cres 5268  cima 5269   Fn wfn 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-fun 6051  df-fn 6052
This theorem is referenced by:  infdifsn  8729  carduniima  9129  cardinfima  9130  alephfp  9141  dprdf1o  18651  dprd2db  18662  lmhmrnlss  19272  mpfsubrg  19754  pf1subrg  19934  frlmlbs  20358  frlmup3  20361  ellspd  20363  tgrest  21185  uniiccdif  23566  uniioombllem3  23573  dvgt0lem2  23985  eulerpartlemn  30773  matunitlindflem2  33737  poimirlem15  33755  k0004lem1  38965
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