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Theorem dff1o2 5131
 Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
dff1o2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))

Proof of Theorem dff1o2
StepHypRef Expression
1 df-f1o 4909 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
2 df-f1 4907 . . . 4 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
3 df-fo 4908 . . . 4 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
42, 3anbi12i 433 . . 3 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ ((𝐹:𝐴𝐵 ∧ Fun 𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
5 anass 381 . . . 4 (((𝐹:𝐴𝐵 ∧ Fun 𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹:𝐴𝐵 ∧ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))))
6 3anan12 897 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
76anbi1i 431 . . . . 5 (((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ∧ 𝐹:𝐴𝐵) ↔ ((Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ 𝐹:𝐴𝐵))
8 eqimss 2997 . . . . . . . 8 (ran 𝐹 = 𝐵 → ran 𝐹𝐵)
9 df-f 4906 . . . . . . . . 9 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
109biimpri 124 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → 𝐹:𝐴𝐵)
118, 10sylan2 270 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴𝐵)
12113adant2 923 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴𝐵)
1312pm4.71i 371 . . . . 5 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ∧ 𝐹:𝐴𝐵))
14 ancom 253 . . . . 5 ((𝐹:𝐴𝐵 ∧ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))) ↔ ((Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ 𝐹:𝐴𝐵))
157, 13, 143bitr4ri 202 . . . 4 ((𝐹:𝐴𝐵 ∧ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
165, 15bitri 173 . . 3 (((𝐹:𝐴𝐵 ∧ Fun 𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
174, 16bitri 173 . 2 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
181, 17bitri 173 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ⊆ wss 2917  ◡ccnv 4344  ran crn 4346  Fun wfun 4896   Fn wfn 4897  ⟶wf 4898  –1-1→wf1 4899  –onto→wfo 4900  –1-1-onto→wf1o 4901 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909 This theorem is referenced by:  dff1o3  5132  dff1o4  5134  f1orn  5136  dif1en  6337
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