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Theorem dff1o4 5165
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 5162 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
2 3anass 924 . 2 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ (Fun 𝐹 ∧ ran 𝐹 = 𝐵)))
3 df-rn 4382 . . . . . 6 ran 𝐹 = dom 𝐹
43eqeq1i 2089 . . . . 5 (ran 𝐹 = 𝐵 ↔ dom 𝐹 = 𝐵)
54anbi2i 445 . . . 4 ((Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐵))
6 df-fn 4935 . . . 4 (𝐹 Fn 𝐵 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐵))
75, 6bitr4i 185 . . 3 ((Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ 𝐹 Fn 𝐵)
87anbi2i 445 . 2 ((𝐹 Fn 𝐴 ∧ (Fun 𝐹 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
91, 2, 83bitri 204 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  w3a 920   = wceq 1285  ccnv 4370  dom cdm 4371  ran crn 4372  Fun wfun 4926   Fn wfn 4927  1-1-ontowf1o 4931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987  df-rn 4382  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939
This theorem is referenced by:  f1ocnv  5170  f1oun  5177  f1o00  5192  f1oi  5195  f1osn  5197  f1ompt  5352  f1ofveu  5531  f1ocnvd  5733  f1od2  5887
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