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 5637	. 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 2783	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)

Syntax hints:   → wi 4   = wceq 1541  ran crn 5625   ↾ cres 5626   “ cima 5627   Fn wfn 6487

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 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-fun 6494  df-fn 6495

This theorem is referenced by:  infdifsn  9566  cardinfima  10007  alephfp  10018  dprdf1o  19963  dprd2db  19974  rnrhmsubrg  20538  lmhmrnlss  21002  frlmlbs  21752  frlmup3  21755  ellspd  21757  mpfsubrg  22066  pf1subrg  22292  tgrest  23103  uniiccdif  25535  uniioombllem3  25542  dvgt0lem2  25964  f1rnen  32706  cycpmco2rn  33207  r1pquslmic  33692  fedgmul  33788  zarclsint  34029  eulerpartlemn  34538  fineqvinfep  35281  matunitlindflem2  37814  poimirlem15  37832  aks6d1c6lem3  42422  aks6d1c6lem5  42427  aks6d1c7lem1  42430  k0004lem1  44384  3f1oss1  47317  imasetpreimafvbijlemf  47643  fundcmpsurbijinjpreimafv  47649