Theorem fnima 6694

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 5669	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6683	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5928	. 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹)
4	1, 3	eqtrid 2810	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ran crn 5657   ↾ cres 5658   “ cima 5659   Fn wfn 6556

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-cnv 5664  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-fun 6563  df-fn 6564

This theorem is used by:  infdifsn  9688  cardinfima  10206  alephfp  10217  dprdf1o  20281  dprd2db  20292  rnrhmsubrg  20890  lmhmrnlss  21358  frlmlbs  22136  frlmup3  22139  ellspd  22141  mpfsubrg  22453  pf1subrg  22699  matunitlindflem2  23028  tgrest  23510  uniiccdif  25932  uniioombllem3  25939  dvgt0lem2  26356  f1rnen  33297  cycpmco2rn  33762  r1pquslmic  34225  fedgmul  34346  zarclsint  34587  eulerpartlemn  35096  fineqvinfep  35906  poimirlem15  38758  aks6d1c6lem3  43425  aks6d1c6lem5  43430  aks6d1c7lem1  43433  k0004lem1  45350  tmachlem-franscan  48174  3f1oss1  48360  imasetpreimafvbijlemf  48698  fundcmpsurbijinjpreimafv  48704