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Theorem nvof1o 7274
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 6646 . . . . . 6 (𝐹 Fn 𝐴 β†’ Fun 𝐹)
2 fdmrn 6746 . . . . . 6 (Fun 𝐹 ↔ 𝐹:dom 𝐹⟢ran 𝐹)
31, 2sylib 217 . . . . 5 (𝐹 Fn 𝐴 β†’ 𝐹:dom 𝐹⟢ran 𝐹)
43adantr 481 . . . 4 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ 𝐹:dom 𝐹⟢ran 𝐹)
5 fndm 6649 . . . . . 6 (𝐹 Fn 𝐴 β†’ dom 𝐹 = 𝐴)
65adantr 481 . . . . 5 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ dom 𝐹 = 𝐴)
7 df-rn 5686 . . . . . . 7 ran 𝐹 = dom ◑𝐹
8 dmeq 5901 . . . . . . 7 (◑𝐹 = 𝐹 β†’ dom ◑𝐹 = dom 𝐹)
97, 8eqtrid 2784 . . . . . 6 (◑𝐹 = 𝐹 β†’ ran 𝐹 = dom 𝐹)
109, 5sylan9eqr 2794 . . . . 5 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ ran 𝐹 = 𝐴)
116, 10feq23d 6709 . . . 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 481 . . . 4 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ Fun 𝐹)
14 funeq 6565 . . . . 5 (◑𝐹 = 𝐹 β†’ (Fun ◑𝐹 ↔ Fun 𝐹))
1514adantl 482 . . . 4 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ (Fun ◑𝐹 ↔ Fun 𝐹))
1613, 15mpbird 256 . . 3 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ Fun ◑𝐹)
17 df-f1 6545 . . 3 (𝐹:𝐴–1-1→𝐴 ↔ (𝐹:𝐴⟢𝐴 ∧ Fun ◑𝐹))
1812, 16, 17sylanbrc 583 . 2 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ 𝐹:𝐴–1-1→𝐴)
19 simpl 483 . . 3 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ 𝐹 Fn 𝐴)
20 df-fo 6546 . . 3 (𝐹:𝐴–onto→𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴))
2119, 10, 20sylanbrc 583 . 2 ((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) β†’ 𝐹:𝐴–onto→𝐴)
22 df-f1o 6547 . 2 (𝐹:𝐴–1-1-onto→𝐴 ↔ (𝐹:𝐴–1-1→𝐴 ∧ 𝐹:𝐴–onto→𝐴))
2318, 21, 22sylanbrc 583 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 396   = wceq 1541  β—‘ccnv 5674  dom cdm 5675  ran crn 5676  Fun wfun 6534   Fn wfn 6535  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€“ontoβ†’wfo 6538  β€“1-1-ontoβ†’wf1o 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547
This theorem is referenced by:  mirf1o  27909  lmif1o  28035  nvocnvb  42158  dssmapf1od  42757
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