Theorem fnima 6651

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 5654	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6640	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5905	. 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹)
4	1, 3	eqtrid 2777	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)

Syntax hints:   → wi 4   = wceq 1540  ran crn 5642   ↾ cres 5643   “ cima 5644   Fn wfn 6509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-fun 6516  df-fn 6517

This theorem is referenced by:  infdifsn  9617  cardinfima  10057  alephfp  10068  dprdf1o  19971  dprd2db  19982  rnrhmsubrg  20521  lmhmrnlss  20964  frlmlbs  21713  frlmup3  21716  ellspd  21718  mpfsubrg  22017  pf1subrg  22242  tgrest  23053  uniiccdif  25486  uniioombllem3  25493  dvgt0lem2  25915  f1rnen  32560  cycpmco2rn  33089  r1pquslmic  33583  fedgmul  33634  zarclsint  33869  eulerpartlemn  34379  matunitlindflem2  37618  poimirlem15  37636  aks6d1c6lem3  42167  aks6d1c6lem5  42172  aks6d1c7lem1  42175  k0004lem1  44143  3f1oss1  47080  imasetpreimafvbijlemf  47406  fundcmpsurbijinjpreimafv  47412