Theorem fnima 6628

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 5644	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6617	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5893	. 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹)
4	1, 3	eqtrid 2783	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)

Syntax hints:   → wi 4   = wceq 1542  ran crn 5632   ↾ cres 5633   “ cima 5634   Fn wfn 6493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6500  df-fn 6501

This theorem is referenced by:  infdifsn  9578  cardinfima  10019  alephfp  10030  dprdf1o  20009  dprd2db  20020  rnrhmsubrg  20582  lmhmrnlss  21045  frlmlbs  21777  frlmup3  21780  ellspd  21782  mpfsubrg  22089  pf1subrg  22313  tgrest  23124  uniiccdif  25545  uniioombllem3  25552  dvgt0lem2  25970  f1rnen  32701  cycpmco2rn  33186  r1pquslmic  33671  fedgmul  33775  zarclsint  34016  eulerpartlemn  34525  fineqvinfep  35269  matunitlindflem2  37938  poimirlem15  37956  aks6d1c6lem3  42611  aks6d1c6lem5  42616  aks6d1c7lem1  42619  k0004lem1  44574  3f1oss1  47523  imasetpreimafvbijlemf  47861  fundcmpsurbijinjpreimafv  47867