Theorem fnima 6648

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 5651	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6637	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5902	. 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 5639   ↾ cres 5640   “ cima 5641   Fn wfn 6506

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 5251  ax-nul 5261  ax-pr 5387

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-fun 6513  df-fn 6514

This theorem is referenced by:  infdifsn  9610  cardinfima  10050  alephfp  10061  dprdf1o  19964  dprd2db  19975  rnrhmsubrg  20514  lmhmrnlss  20957  frlmlbs  21706  frlmup3  21709  ellspd  21711  mpfsubrg  22010  pf1subrg  22235  tgrest  23046  uniiccdif  25479  uniioombllem3  25486  dvgt0lem2  25908  f1rnen  32553  cycpmco2rn  33082  r1pquslmic  33576  fedgmul  33627  zarclsint  33862  eulerpartlemn  34372  matunitlindflem2  37611  poimirlem15  37629  aks6d1c6lem3  42160  aks6d1c6lem5  42165  aks6d1c7lem1  42168  k0004lem1  44136  3f1oss1  47076  imasetpreimafvbijlemf  47402  fundcmpsurbijinjpreimafv  47408