Theorem fnima 6630

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 5645	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6619	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5895	. 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 5633   ↾ cres 5634   “ cima 5635   Fn wfn 6495

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 5243  ax-pr 5379

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 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-fun 6502  df-fn 6503

This theorem is referenced by:  infdifsn  9578  cardinfima  10019  alephfp  10030  dprdf1o  19975  dprd2db  19986  rnrhmsubrg  20550  lmhmrnlss  21014  frlmlbs  21764  frlmup3  21767  ellspd  21769  mpfsubrg  22078  pf1subrg  22304  tgrest  23115  uniiccdif  25547  uniioombllem3  25554  dvgt0lem2  25976  f1rnen  32717  cycpmco2rn  33218  r1pquslmic  33703  fedgmul  33808  zarclsint  34049  eulerpartlemn  34558  fineqvinfep  35300  matunitlindflem2  37862  poimirlem15  37880  aks6d1c6lem3  42536  aks6d1c6lem5  42541  aks6d1c7lem1  42544  k0004lem1  44497  3f1oss1  47429  imasetpreimafvbijlemf  47755  fundcmpsurbijinjpreimafv  47761