MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnima Structured version   Visualization version   GIF version

Theorem fnima 5905
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 5037 . 2 (𝐹𝐴) = ran (𝐹𝐴)
2 fnresdm 5896 . . 3 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
32rneqd 5257 . 2 (𝐹 Fn 𝐴 → ran (𝐹𝐴) = ran 𝐹)
41, 3syl5eq 2651 1 (𝐹 Fn 𝐴 → (𝐹𝐴) = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  ran crn 5025  cres 5026  cima 5027   Fn wfn 5781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-opab 4634  df-xp 5030  df-rel 5031  df-cnv 5032  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-fun 5788  df-fn 5789
This theorem is referenced by:  infdifsn  8410  carduniima  8775  cardinfima  8776  alephfp  8787  dprdf1o  18196  dprd2db  18207  lmhmrnlss  18813  mpfsubrg  19295  pf1subrg  19475  frlmlbs  19893  frlmup3  19896  ellspd  19898  tgrest  20711  uniiccdif  23065  uniioombllem3  23072  dvgt0lem2  23483  eulerpartlemn  29572  matunitlindflem2  32375  poimirlem15  32393  k0004lem1  37264
  Copyright terms: Public domain W3C validator