Theorem f1f1orn 5582

Description: A one-to-one function maps one-to-one onto its range. (Contributed by NM, 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 5532	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		df-f1 5322	. . 3 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ◡𝐹))
3	2	simprbi 275	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun ◡𝐹)
4		f1orn 5581	. 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹))
5	1, 3, 4	sylanbrc 417	1 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)

Syntax hints:   → wi 4  ^◡ccnv 4717  ran crn 4719  Fun wfun 5311   Fn wfn 5312  ⟶wf 5313  –1-1→wf1 5314  –1-1-onto→wf1o 5316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324

This theorem is referenced by:  f1ores  5586  f1cnv  5595  f1cocnv1  5601  f1ocnvfvrneq  5905  ssenen  7008  f1dmvrnfibi  7107  cc2lem  7448  4sqlem11  12919  xpsff1o2  13379  imasmndf1  13482  imasgrpf1  13644  conjsubgen  13810  imasrngf1  13915  imasringf1  14023  usgrf1o  15966  uspgrf1oedg  15968