Theorem fnima 6668

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 5667	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6657	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5918	. 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹)
4	1, 3	eqtrid 2782	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)

Syntax hints:   → wi 4   = wceq 1540  ran crn 5655   ↾ cres 5656   “ cima 5657   Fn wfn 6526

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-fun 6533  df-fn 6534

This theorem is referenced by:  infdifsn  9671  cardinfima  10111  alephfp  10122  dprdf1o  20015  dprd2db  20026  rnrhmsubrg  20565  lmhmrnlss  21008  frlmlbs  21757  frlmup3  21760  ellspd  21762  mpfsubrg  22061  pf1subrg  22286  tgrest  23097  uniiccdif  25531  uniioombllem3  25538  dvgt0lem2  25960  f1rnen  32607  cycpmco2rn  33136  r1pquslmic  33620  fedgmul  33671  zarclsint  33903  eulerpartlemn  34413  matunitlindflem2  37641  poimirlem15  37659  aks6d1c6lem3  42185  aks6d1c6lem5  42190  aks6d1c7lem1  42193  k0004lem1  44171  3f1oss1  47104  imasetpreimafvbijlemf  47415  fundcmpsurbijinjpreimafv  47421