Theorem fnima 6622

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 5638	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6611	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5887	. 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹)
4	1, 3	eqtrid 2787	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)

Syntax hints:   → wi 4   = wceq 1547  ran crn 5626   ↾ cres 5627   “ cima 5628   Fn wfn 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-fun 6494  df-fn 6495

This theorem is referenced by:  infdifsn  9576  cardinfima  10017  alephfp  10028  dprdf1o  20007  dprd2db  20018  rnrhmsubrg  20584  lmhmrnlss  21047  frlmlbs  21779  frlmup3  21782  ellspd  21784  mpfsubrg  22094  pf1subrg  22341  tgrest  23149  uniiccdif  25570  uniioombllem3  25577  dvgt0lem2  25995  f1rnen  32727  cycpmco2rn  33213  r1pquslmic  33701  fedgmul  33822  zarclsint  34063  eulerpartlemn  34572  fineqvinfep  35313  matunitlindflem2  37991  poimirlem15  38009  aks6d1c6lem3  42664  aks6d1c6lem5  42669  aks6d1c7lem1  42672  k0004lem1  44598  3f1oss1  47545  imasetpreimafvbijlemf  47883  fundcmpsurbijinjpreimafv  47889