Theorem fnima 6616

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 5636	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6605	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5884	. 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹)
4	1, 3	eqtrid 2776	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)

Syntax hints:   → wi 4   = wceq 1540  ran crn 5624   ↾ cres 5625   “ cima 5626   Fn wfn 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-fun 6488  df-fn 6489

This theorem is referenced by:  infdifsn  9572  cardinfima  10010  alephfp  10021  dprdf1o  19931  dprd2db  19942  rnrhmsubrg  20508  lmhmrnlss  20972  frlmlbs  21722  frlmup3  21725  ellspd  21727  mpfsubrg  22026  pf1subrg  22251  tgrest  23062  uniiccdif  25495  uniioombllem3  25502  dvgt0lem2  25924  f1rnen  32586  cycpmco2rn  33080  r1pquslmic  33552  fedgmul  33603  zarclsint  33838  eulerpartlemn  34348  matunitlindflem2  37596  poimirlem15  37614  aks6d1c6lem3  42145  aks6d1c6lem5  42150  aks6d1c7lem1  42153  k0004lem1  44120  3f1oss1  47060  imasetpreimafvbijlemf  47386  fundcmpsurbijinjpreimafv  47392