Theorem fnima 6611

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

Step	Hyp	Ref	Expression
1		df-ima 5629	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6600	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5878	. 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹)
4	1, 3	eqtrid 2778	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)

Syntax hints:   → wi 4   = wceq 1541  ran crn 5617   ↾ cres 5618   “ cima 5619   Fn wfn 6476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-fun 6483  df-fn 6484

This theorem is referenced by:  infdifsn  9547  cardinfima  9985  alephfp  9996  dprdf1o  19944  dprd2db  19955  rnrhmsubrg  20518  lmhmrnlss  20982  frlmlbs  21732  frlmup3  21735  ellspd  21737  mpfsubrg  22036  pf1subrg  22261  tgrest  23072  uniiccdif  25504  uniioombllem3  25511  dvgt0lem2  25933  f1rnen  32605  cycpmco2rn  33089  r1pquslmic  33566  fedgmul  33639  zarclsint  33880  eulerpartlemn  34389  matunitlindflem2  37656  poimirlem15  37674  aks6d1c6lem3  42204  aks6d1c6lem5  42209  aks6d1c7lem1  42212  k0004lem1  44179  3f1oss1  47105  imasetpreimafvbijlemf  47431  fundcmpsurbijinjpreimafv  47437