Theorem f1f1orn 5594

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 5544	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		df-f1 5331	. . 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 5593	. 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 4724  ran crn 4726  Fun wfun 5320   Fn wfn 5321  ⟶wf 5322  –1-1→wf1 5323  –1-1-onto→wf1o 5325

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333

This theorem is referenced by:  f1ores  5598  f1cnv  5607  f1cocnv1  5613  f1ocnvfvrneq  5923  ssenen  7037  f1dmvrnfibi  7143  cc2lem  7485  4sqlem11  12979  xpsff1o2  13439  imasmndf1  13542  imasgrpf1  13704  conjsubgen  13870  imasrngf1  13976  imasringf1  14084  usgrf1o  16031  uspgrf1oedg  16033