Theorem fnima 6655

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 5665	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6644	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5919	. 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹)
4	1, 3	eqtrid 2812	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)

Syntax hints:   → wi 4   = wceq 1563  ran crn 5653   ↾ cres 5654   “ cima 5655   Fn wfn 6520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-fun 6527  df-fn 6528

This theorem is referenced by:  infdifsn  9614  cardinfima  10069  alephfp  10080  dprdf1o  20095  dprd2db  20106  rnrhmsubrg  20681  lmhmrnlss  21140  frlmlbs  21907  frlmup3  21910  ellspd  21912  mpfsubrg  22222  pf1subrg  22469  tgrest  23277  uniiccdif  25698  uniioombllem3  25705  dvgt0lem2  26123  f1rnen  32885  cycpmco2rn  33358  r1pquslmic  33817  fedgmul  33938  zarclsint  34179  eulerpartlemn  34688  fineqvinfep  35433  matunitlindflem2  38128  poimirlem15  38146  aks6d1c6lem3  42801  aks6d1c6lem5  42806  aks6d1c7lem1  42809  k0004lem1  44735  3f1oss1  47667  imasetpreimafvbijlemf  48005  fundcmpsurbijinjpreimafv  48011