ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1f1orn GIF version

Theorem f1f1orn 5248
Description: A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)
Assertion
Ref Expression
f1f1orn (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)

Proof of Theorem f1f1orn
StepHypRef Expression
1 f1fn 5202 . 2 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
2 df-f1 5007 . . 3 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
32simprbi 269 . 2 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
4 f1orn 5247 . 2 (𝐹:𝐴1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))
51, 3, 4sylanbrc 408 1 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  ccnv 4427  ran crn 4429  Fun wfun 4996   Fn wfn 4997  wf 4998  1-1wf1 4999  1-1-ontowf1o 5001
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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009
This theorem is referenced by:  f1ores  5252  f1cnv  5261  f1cocnv1  5267  f1ocnvfvrneq  5543  ssenen  6547  f1dmvrnfibi  6632
  Copyright terms: Public domain W3C validator