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 2784	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)

Syntax hints:   → wi 4   = wceq 1542  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 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  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  9569  cardinfima  10010  alephfp  10021  dprdf1o  20000  dprd2db  20011  rnrhmsubrg  20573  lmhmrnlss  21037  frlmlbs  21787  frlmup3  21790  ellspd  21792  mpfsubrg  22099  pf1subrg  22323  tgrest  23134  uniiccdif  25555  uniioombllem3  25562  dvgt0lem2  25980  f1rnen  32716  cycpmco2rn  33201  r1pquslmic  33686  fedgmul  33791  zarclsint  34032  eulerpartlemn  34541  fineqvinfep  35285  matunitlindflem2  37952  poimirlem15  37970  aks6d1c6lem3  42625  aks6d1c6lem5  42630  aks6d1c7lem1  42633  k0004lem1  44592  3f1oss1  47535  imasetpreimafvbijlemf  47873  fundcmpsurbijinjpreimafv  47879