Theorem fnima 6699

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 5702	. 2 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
2		fnresdm 6688	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
3	2	rneqd 5952	. 2 ⊢ (𝐹 Fn 𝐴 → ran (𝐹 ↾ 𝐴) = ran 𝐹)
4	1, 3	eqtrid 2787	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)

Syntax hints:   → wi 4   = wceq 1537  ran crn 5690   ↾ cres 5691   “ cima 5692   Fn wfn 6558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566

This theorem is referenced by:  infdifsn  9695  cardinfima  10135  alephfp  10146  dprdf1o  20067  dprd2db  20078  rnrhmsubrg  20622  lmhmrnlss  21067  frlmlbs  21835  frlmup3  21838  ellspd  21840  mpfsubrg  22145  pf1subrg  22368  tgrest  23183  uniiccdif  25627  uniioombllem3  25634  dvgt0lem2  26057  f1rnen  32646  cycpmco2rn  33128  r1pquslmic  33611  fedgmul  33659  zarclsint  33833  eulerpartlemn  34363  matunitlindflem2  37604  poimirlem15  37622  aks6d1c6lem3  42154  aks6d1c6lem5  42159  aks6d1c7lem1  42162  k0004lem1  44137  3f1oss1  47025  imasetpreimafvbijlemf  47326  fundcmpsurbijinjpreimafv  47332