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Theorem f1orn 5163
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 5158 . 2 (𝐹:𝐴1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = ran 𝐹))
2 eqid 2056 . . 3 ran 𝐹 = ran 𝐹
3 df-3an 898 . . 3 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ ((𝐹 Fn 𝐴 ∧ Fun 𝐹) ∧ ran 𝐹 = ran 𝐹))
42, 3mpbiran2 859 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))
51, 4bitri 177 1 (𝐹:𝐴1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  w3a 896   = wceq 1259  ccnv 4371  ran crn 4373  Fun wfun 4923   Fn wfn 4924  1-1-ontowf1o 4928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2951  df-ss 2958  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936
This theorem is referenced by:  f1f1orn  5164
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