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Theorem f1f1orn 5297
 Description: A one-to-one function maps one-to-one onto its range. (Contributed by set.mm contributors, 4-Sep-2004.)
Assertion
Ref Expression
f1f1orn (F:A1-1BF:A1-1-onto→ran F)

Proof of Theorem f1f1orn
StepHypRef Expression
1 f1fn 5259 . 2 (F:A1-1BF Fn A)
2 df-f1 4792 . . 3 (F:A1-1B ↔ (F:A–→B Fun F))
32simprbi 450 . 2 (F:A1-1B → Fun F)
4 f1orn 5296 . 2 (F:A1-1-onto→ran F ↔ (F Fn A Fun F))
51, 3, 4sylanbrc 645 1 (F:A1-1BF:A1-1-onto→ran F)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ◡ccnv 4771  ran crn 4773  Fun wfun 4775   Fn wfn 4776  –→wf 4777  –1-1→wf1 4778  –1-1-onto→wf1o 4780 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794 This theorem is referenced by:  f1cnv  5311  f1cocnv1  5312  f1cocnv2  5313  dflec3  6221
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