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Theorem f1orn 5467
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 5462 . 2 (𝐹:𝐴1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = ran 𝐹))
2 eqid 2177 . . 3 ran 𝐹 = ran 𝐹
3 df-3an 980 . . 3 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ ((𝐹 Fn 𝐴 ∧ Fun 𝐹) ∧ ran 𝐹 = ran 𝐹))
42, 3mpbiran2 941 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))
51, 4bitri 184 1 (𝐹:𝐴1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  w3a 978   = wceq 1353  ccnv 4622  ran crn 4624  Fun wfun 5206   Fn wfn 5207  1-1-ontowf1o 5211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219
This theorem is referenced by:  f1f1orn  5468
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