Theorem f1f1orn 5603

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 5553	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		df-f1 5338	. . 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 5602	. 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 4730  ran crn 4732  Fun wfun 5327   Fn wfn 5328  ⟶wf 5329  –1-1→wf1 5330  –1-1-onto→wf1o 5332

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3207  df-ss 3214  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340

This theorem is referenced by:  f1ores  5607  f1cnv  5616  f1cocnv1  5622  f1ocnvfvrneq  5933  ssenen  7080  f1dmvrnfibi  7186  cc2lem  7528  4sqlem11  13037  xpsff1o2  13497  imasmndf1  13600  imasgrpf1  13762  conjsubgen  13928  imasrngf1  14034  imasringf1  14142  usgrf1o  16098  uspgrf1oedg  16100