Theorem fnima 6669

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 5663	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6658	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5919	. 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹)
4	1, 3	eqtrid 2807	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ran crn 5651   ↾ cres 5652   “ cima 5653   Fn wfn 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-fun 6540  df-fn 6541

This theorem is used by:  infdifsn  9661  cardinfima  10179  alephfp  10190  dprdf1o  20253  dprd2db  20264  rnrhmsubrg  20862  lmhmrnlss  21330  frlmlbs  22108  frlmup3  22111  ellspd  22113  mpfsubrg  22425  pf1subrg  22671  matunitlindflem2  23000  tgrest  23482  uniiccdif  25904  uniioombllem3  25911  dvgt0lem2  26328  f1rnen  33269  cycpmco2rn  33732  r1pquslmic  34188  fedgmul  34309  zarclsint  34550  eulerpartlemn  35059  fineqvinfep  35849  poimirlem15  38646  aks6d1c6lem3  43315  aks6d1c6lem5  43320  aks6d1c7lem1  43323  k0004lem1  45240  tmachlem-franscan  48052  3f1oss1  48238  imasetpreimafvbijlemf  48576  fundcmpsurbijinjpreimafv  48582