Theorem fnima 6672

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 5679	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6661	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5933	. 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹)
4	1, 3	eqtrid 2813	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ran crn 5667   ↾ cres 5668   “ cima 5669   Fn wfn 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-fun 6545  df-fn 6546

This theorem is used by:  infdifsn  9636  cardinfima  10100  alephfp  10111  dprdf1o  20135  dprd2db  20146  rnrhmsubrg  20741  lmhmrnlss  21208  frlmlbs  21984  frlmup3  21987  ellspd  21989  mpfsubrg  22299  pf1subrg  22545  tgrest  23353  uniiccdif  25774  uniioombllem3  25781  dvgt0lem2  26199  f1rnen  33010  cycpmco2rn  33476  r1pquslmic  33931  fedgmul  34052  zarclsint  34293  eulerpartlemn  34803  fineqvinfep  35562  matunitlindflem2  38309  poimirlem15  38327  aks6d1c6lem3  42980  aks6d1c6lem5  42985  aks6d1c7lem1  42988  k0004lem1  44914  3f1oss1  47853  imasetpreimafvbijlemf  48191  fundcmpsurbijinjpreimafv  48197