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Theorem nvof1o 7278
Description: An involution is a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
Assertion
Ref Expression
nvof1o ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ 𝐹:𝐴–1-1-onto→𝐴)

Proof of Theorem nvof1o
StepHypRef Expression
1 fnfun 6650 . . . . . 6 (𝐹 Fn 𝐴 β†’ Fun 𝐹)
2 fdmrn 6750 . . . . . 6 (Fun 𝐹 ↔ 𝐹:dom 𝐹⟢ran 𝐹)
31, 2sylib 217 . . . . 5 (𝐹 Fn 𝐴 β†’ 𝐹:dom 𝐹⟢ran 𝐹)
43adantr 482 . . . 4 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ 𝐹:dom 𝐹⟢ran 𝐹)
5 fndm 6653 . . . . . 6 (𝐹 Fn 𝐴 β†’ dom 𝐹 = 𝐴)
65adantr 482 . . . . 5 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ dom 𝐹 = 𝐴)
7 df-rn 5688 . . . . . . 7 ran 𝐹 = dom ◑𝐹
8 dmeq 5904 . . . . . . 7 (◑𝐹 = 𝐹 β†’ dom ◑𝐹 = dom 𝐹)
97, 8eqtrid 2785 . . . . . 6 (◑𝐹 = 𝐹 β†’ ran 𝐹 = dom 𝐹)
109, 5sylan9eqr 2795 . . . . 5 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ ran 𝐹 = 𝐴)
116, 10feq23d 6713 . . . 4 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ (𝐹:dom 𝐹⟢ran 𝐹 ↔ 𝐹:𝐴⟢𝐴))
124, 11mpbid 231 . . 3 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ 𝐹:𝐴⟢𝐴)
131adantr 482 . . . 4 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ Fun 𝐹)
14 funeq 6569 . . . . 5 (◑𝐹 = 𝐹 β†’ (Fun ◑𝐹 ↔ Fun 𝐹))
1514adantl 483 . . . 4 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ (Fun ◑𝐹 ↔ Fun 𝐹))
1613, 15mpbird 257 . . 3 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ Fun ◑𝐹)
17 df-f1 6549 . . 3 (𝐹:𝐴–1-1→𝐴 ↔ (𝐹:𝐴⟢𝐴 ∧ Fun ◑𝐹))
1812, 16, 17sylanbrc 584 . 2 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ 𝐹:𝐴–1-1→𝐴)
19 simpl 484 . . 3 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ 𝐹 Fn 𝐴)
20 df-fo 6550 . . 3 (𝐹:𝐴–onto→𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴))
2119, 10, 20sylanbrc 584 . 2 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ 𝐹:𝐴–onto→𝐴)
22 df-f1o 6551 . 2 (𝐹:𝐴–1-1-onto→𝐴 ↔ (𝐹:𝐴–1-1→𝐴 ∧ 𝐹:𝐴–onto→𝐴))
2318, 21, 22sylanbrc 584 1 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ 𝐹:𝐴–1-1-onto→𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  β—‘ccnv 5676  dom cdm 5677  ran crn 5678  Fun wfun 6538   Fn wfn 6539  βŸΆwf 6540  β€“1-1β†’wf1 6541  β€“ontoβ†’wfo 6542  β€“1-1-ontoβ†’wf1o 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551
This theorem is referenced by:  mirf1o  27920  lmif1o  28046  nvocnvb  42173  dssmapf1od  42772
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