Theorem f1f1orn 5650

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 5600	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		df-f1 5382	. . 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 5649	. 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹))
5	1, 3, 4	sylanbrc 421	1 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)

This proof depends on syntax axioms:   → wi 4  ^◡ccnv 4773  ran crn 4775  Fun wfun 5371   Fn wfn 5372  ⟶wf 5373  –1-1→wf1 5374  –1-1-onto→wf1o 5376

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384

This theorem is used by:  f1ores  5654  f1cnv  5663  f1cocnv1  5669  f1ocnvfvrneq  5988  ssenen  7152  f1dmvrnfibi  7258  cc2lem  7632  hashf1lem1  11285  hashf1lem2  11286  4sqlem11  13180  xpsff1o2  13672  imasmndf1  13761  imasgrpf1  13915  conjsubgen  14081  imasrngf1  14256  imasringf1  14370  usgrf1o  16415  uspgrf1oedg  16417