Theorem fnima 6698

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 5698	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6687	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5949	. 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹)
4	1, 3	eqtrid 2789	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)

Syntax hints:   → wi 4   = wceq 1540  ran crn 5686   ↾ cres 5687   “ cima 5688   Fn wfn 6556

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564

This theorem is referenced by:  infdifsn  9697  cardinfima  10137  alephfp  10148  dprdf1o  20052  dprd2db  20063  rnrhmsubrg  20605  lmhmrnlss  21049  frlmlbs  21817  frlmup3  21820  ellspd  21822  mpfsubrg  22127  pf1subrg  22352  tgrest  23167  uniiccdif  25613  uniioombllem3  25620  dvgt0lem2  26042  f1rnen  32639  cycpmco2rn  33145  r1pquslmic  33631  fedgmul  33682  zarclsint  33871  eulerpartlemn  34383  matunitlindflem2  37624  poimirlem15  37642  aks6d1c6lem3  42173  aks6d1c6lem5  42178  aks6d1c7lem1  42181  k0004lem1  44160  3f1oss1  47087  imasetpreimafvbijlemf  47388  fundcmpsurbijinjpreimafv  47394