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Theorem dff1o2 6613
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 6355 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵))
2 df-f1 6353 . . . 4 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
3 df-fo 6354 . . . 4 (𝐹:𝐴onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
42, 3anbi12i 626 . . 3 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ ((𝐹:𝐴𝐵 ∧ Fun 𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
5 anass 469 . . . 4 (((𝐹:𝐴𝐵 ∧ Fun 𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹:𝐴𝐵 ∧ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))))
6 3anan12 1088 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)))
76anbi1i 623 . . . . 5 (((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ∧ 𝐹:𝐴𝐵) ↔ ((Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ 𝐹:𝐴𝐵))
8 eqimss 4020 . . . . . . . 8 (ran 𝐹 = 𝐵 → ran 𝐹𝐵)
9 df-f 6352 . . . . . . . . 9 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
109biimpri 229 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → 𝐹:𝐴𝐵)
118, 10sylan2 592 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴𝐵)
12113adant2 1123 . . . . . 6 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) → 𝐹:𝐴𝐵)
1312pm4.71i 560 . . . . 5 ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ↔ ((𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵) ∧ 𝐹:𝐴𝐵))
14 ancom 461 . . . . 5 ((𝐹:𝐴𝐵 ∧ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))) ↔ ((Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ∧ 𝐹:𝐴𝐵))
157, 13, 143bitr4ri 305 . . . 4 ((𝐹:𝐴𝐵 ∧ (Fun 𝐹 ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
165, 15bitri 276 . . 3 (((𝐹:𝐴𝐵 ∧ Fun 𝐹) ∧ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
174, 16bitri 276 . 2 ((𝐹:𝐴1-1𝐵𝐹:𝐴onto𝐵) ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
181, 17bitri 276 1 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1079   = wceq 1528  wss 3933  ccnv 5547  ran crn 5549  Fun wfun 6342   Fn wfn 6343  wf 6344  1-1wf1 6345  ontowfo 6346  1-1-ontowf1o 6347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-in 3940  df-ss 3949  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355
This theorem is referenced by:  dff1o3  6614  dff1o4  6616  f1orn  6618  tz7.49c  8071  fiint  8783  symgfixelsi  18492  dfrelog  25076  adj1o  29598  fresf1o  30304  f1mptrn  30308  esumc  31209  ntrneinex  40305  stoweidlem39  42201
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