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Theorem dff1o2 5242
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 5009 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
2 df-f1 5007 . . . 4 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
3 df-fo 5008 . . . 4 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
42, 3anbi12i 448 . . 3 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ ((𝐹:𝐴𝐵 ∧ Fun 𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
5 anass 393 . . . 4 (((𝐹:𝐴𝐵 ∧ Fun 𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹:𝐴𝐵 ∧ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))))
6 3anan12 936 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
76anbi1i 446 . . . . 5 (((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ∧ 𝐹:𝐴𝐵) ↔ ((Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ 𝐹:𝐴𝐵))
8 eqimss 3076 . . . . . . . 8 (ran 𝐹 = 𝐵 → ran 𝐹𝐵)
9 df-f 5006 . . . . . . . . 9 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
109biimpri 131 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → 𝐹:𝐴𝐵)
118, 10sylan2 280 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴𝐵)
12113adant2 962 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴𝐵)
1312pm4.71i 383 . . . . 5 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ∧ 𝐹:𝐴𝐵))
14 ancom 262 . . . . 5 ((𝐹:𝐴𝐵 ∧ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))) ↔ ((Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ 𝐹:𝐴𝐵))
157, 13, 143bitr4ri 211 . . . 4 ((𝐹:𝐴𝐵 ∧ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
165, 15bitri 182 . . 3 (((𝐹:𝐴𝐵 ∧ Fun 𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
174, 16bitri 182 . 2 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
181, 17bitri 182 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  w3a 924   = wceq 1289  wss 2997  ccnv 4427  ran crn 4429  Fun wfun 4996   Fn wfn 4997  wf 4998  1-1wf1 4999  ontowfo 5000  1-1-ontowf1o 5001
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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009
This theorem is referenced by:  dff1o3  5243  dff1o4  5245  f1orn  5247  dif1en  6575
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