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Theorem fnima 6008
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 → (𝐹𝐴) = ran 𝐹)

Proof of Theorem fnima
StepHypRef Expression
1 df-ima 5125 . 2 (𝐹𝐴) = ran (𝐹𝐴)
2 fnresdm 5998 . . 3 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
32rneqd 5351 . 2 (𝐹 Fn 𝐴 → ran (𝐹𝐴) = ran 𝐹)
41, 3syl5eq 2667 1 (𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  ran crn 5113  cres 5114  cima 5115   Fn wfn 5881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-fun 5888  df-fn 5889
This theorem is referenced by:  infdifsn  8551  carduniima  8916  cardinfima  8917  alephfp  8928  dprdf1o  18425  dprd2db  18436  lmhmrnlss  19044  mpfsubrg  19526  pf1subrg  19706  frlmlbs  20130  frlmup3  20133  ellspd  20135  tgrest  20957  uniiccdif  23340  uniioombllem3  23347  dvgt0lem2  23760  eulerpartlemn  30428  matunitlindflem2  33386  poimirlem15  33404  k0004lem1  38271
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