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Theorem f1orn 5136
 Description: A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
f1orn (𝐹:𝐴1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))

Proof of Theorem f1orn
StepHypRef Expression
1 dff1o2 5131 . 2 (𝐹:𝐴1-1-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))
42, 3mpbiran2 848 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))
51, 4bitri 173 1 (𝐹:𝐴1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))
 Colors of variables: wff set class 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
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