Theorem f1orn 5136

Description: A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)

Assertion

Ref	Expression
f1orn	⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹))

Proof of Theorem f1orn

Step	Hyp	Ref	Expression
1		dff1o2 5131	. 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹))
2		eqid 2040	. . 3 ⊢ ran 𝐹 = ran 𝐹
3		df-3an 887	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹) ∧ ran 𝐹 = ran 𝐹))
4	2, 3	mpbiran2 848	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹))
5	1, 4	bitri 173	1 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹))

Syntax hints:   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243  ^◡ccnv 4344  ran crn 4346  Fun wfun 4896   Fn wfn 4897  –1-1-onto→wf1o 4901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909

This theorem is referenced by:  f1f1orn  5137