Theorem fnima 6647

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 5658	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6636	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5912	. 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹)
4	1, 3	eqtrid 2808	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)

Syntax hints:   → wi 4   = wceq 1559  ran crn 5646   ↾ cres 5647   “ cima 5648   Fn wfn 6512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-fun 6519  df-fn 6520

This theorem is referenced by:  infdifsn  9609  cardinfima  10050  alephfp  10061  dprdf1o  20057  dprd2db  20068  rnrhmsubrg  20634  lmhmrnlss  21097  frlmlbs  21829  frlmup3  21832  ellspd  21834  mpfsubrg  22144  pf1subrg  22391  tgrest  23199  uniiccdif  25620  uniioombllem3  25627  dvgt0lem2  26045  f1rnen  32780  cycpmco2rn  33266  r1pquslmic  33768  fedgmul  33889  zarclsint  34130  eulerpartlemn  34639  fineqvinfep  35385  matunitlindflem2  38080  poimirlem15  38098  aks6d1c6lem3  42753  aks6d1c6lem5  42758  aks6d1c7lem1  42761  k0004lem1  44687  3f1oss1  47633  imasetpreimafvbijlemf  47971  fundcmpsurbijinjpreimafv  47977