Theorem fnima 6632

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 5647	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6621	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5897	. 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 5635   ↾ cres 5636   “ cima 5637   Fn wfn 6497

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 5245  ax-pr 5381

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 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6504  df-fn 6505

This theorem is referenced by:  infdifsn  9580  cardinfima  10021  alephfp  10032  dprdf1o  19980  dprd2db  19991  rnrhmsubrg  20555  lmhmrnlss  21019  frlmlbs  21769  frlmup3  21772  ellspd  21774  mpfsubrg  22083  pf1subrg  22309  tgrest  23120  uniiccdif  25552  uniioombllem3  25559  dvgt0lem2  25981  f1rnen  32724  cycpmco2rn  33225  r1pquslmic  33710  fedgmul  33815  zarclsint  34056  eulerpartlemn  34565  fineqvinfep  35309  matunitlindflem2  37897  poimirlem15  37915  aks6d1c6lem3  42571  aks6d1c6lem5  42576  aks6d1c7lem1  42579  k0004lem1  44532  3f1oss1  47464  imasetpreimafvbijlemf  47790  fundcmpsurbijinjpreimafv  47796