ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  foima GIF version

Theorem foima 5238
Description: The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
Assertion
Ref Expression
foima (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)

Proof of Theorem foima
StepHypRef Expression
1 imadmrn 4784 . 2 (𝐹 “ dom 𝐹) = ran 𝐹
2 fof 5233 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
3 fdm 5166 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
42, 3syl 14 . . 3 (𝐹:𝐴onto𝐵 → dom 𝐹 = 𝐴)
54imaeq2d 4774 . 2 (𝐹:𝐴onto𝐵 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
6 forn 5236 . 2 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
71, 5, 63eqtr3a 2144 1 (𝐹:𝐴onto𝐵 → (𝐹𝐴) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  dom cdm 4438  ran crn 4439  cima 4441  wf 5011  ontowfo 5013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-fn 5018  df-f 5019  df-fo 5021
This theorem is referenced by:  foimacnv  5271  foima2  5530  fiintim  6639  fidcenumlemr  6664
  Copyright terms: Public domain W3C validator