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Theorem dff1o4 5419
 Description: Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
dff1o4 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))

Proof of Theorem dff1o4
StepHypRef Expression
1 dff1o2 5416 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
2 3anass 967 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ (Fun 𝐹 ∧ ran 𝐹 = 𝐵)))
3 df-rn 4594 . . . . . 6 ran 𝐹 = dom 𝐹
43eqeq1i 2165 . . . . 5 (ran 𝐹 = 𝐵 ↔ dom 𝐹 = 𝐵)
54anbi2i 453 . . . 4 ((Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐵))
6 df-fn 5170 . . . 4 (𝐹 Fn 𝐵 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐵))
75, 6bitr4i 186 . . 3 ((Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ 𝐹 Fn 𝐵)
87anbi2i 453 . 2 ((𝐹 Fn 𝐴 ∧ (Fun 𝐹 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
91, 2, 83bitri 205 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1335  ◡ccnv 4582  dom cdm 4583  ran crn 4584  Fun wfun 5161   Fn wfn 5162  –1-1-onto→wf1o 5166 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115  df-rn 4594  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174 This theorem is referenced by:  f1ocnv  5424  f1oun  5431  f1o00  5446  f1oi  5449  f1osn  5451  f1ompt  5615  f1ofveu  5806  f1ocnvd  6016  f1od2  6176  mapsnf1o2  6634  sbthlemi9  6902  hmeof1o2  12668
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