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Theorem f1orn 6300
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 6295 . 2 (𝐹:𝐴1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = ran 𝐹))
2 eqid 2752 . . 3 ran 𝐹 = ran 𝐹
3 df-3an 1074 . . 3 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ ((𝐹 Fn 𝐴 ∧ Fun 𝐹) ∧ ran 𝐹 = ran 𝐹))
42, 3mpbiran2 992 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))
51, 4bitri 264 1 (𝐹:𝐴1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1624  ccnv 5257  ran crn 5259  Fun wfun 6035   Fn wfn 6036  1-1-ontowf1o 6040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-in 3714  df-ss 3721  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048
This theorem is referenced by:  f1f1orn  6301  infdifsn  8719  efopnlem2  24594
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