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:A–1-1→B → F:A–1-1-onto→ran F)

Proof of Theorem f1f1orn

Step	Hyp	Ref	Expression
1		f1fn 5259	. 2 ⊢ (F:A–1-1→B → F Fn A)
2		df-f1 4792	. . 3 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ Fun ◡F))
3	2	simprbi 450	. 2 ⊢ (F:A–1-1→B → Fun ◡F)
4		f1orn 5296	. 2 ⊢ (F:A–1-1-onto→ran F ↔ (F Fn A ∧ Fun ◡F))
5	1, 3, 4	sylanbrc 645	1 ⊢ (F:A–1-1→B → F:A–1-1-onto→ran F)

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-mp 5  ax-1 6  ax-2 7  ax-3 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