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Theorem nvof1o 7300
Description: An involution is a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
Assertion
Ref Expression
nvof1o ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1-onto𝐴)

Proof of Theorem nvof1o
StepHypRef Expression
1 fnfun 6668 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fdmrn 6767 . . . . . 6 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
31, 2sylib 218 . . . . 5 (𝐹 Fn 𝐴𝐹:dom 𝐹⟶ran 𝐹)
43adantr 480 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:dom 𝐹⟶ran 𝐹)
5 fndm 6671 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65adantr 480 . . . . 5 ((𝐹 Fn 𝐴𝐹 = 𝐹) → dom 𝐹 = 𝐴)
7 df-rn 5696 . . . . . . 7 ran 𝐹 = dom 𝐹
8 dmeq 5914 . . . . . . 7 (𝐹 = 𝐹 → dom 𝐹 = dom 𝐹)
97, 8eqtrid 2789 . . . . . 6 (𝐹 = 𝐹 → ran 𝐹 = dom 𝐹)
109, 5sylan9eqr 2799 . . . . 5 ((𝐹 Fn 𝐴𝐹 = 𝐹) → ran 𝐹 = 𝐴)
116, 10feq23d 6731 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → (𝐹:dom 𝐹⟶ran 𝐹𝐹:𝐴𝐴))
124, 11mpbid 232 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴𝐴)
131adantr 480 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → Fun 𝐹)
14 funeq 6586 . . . . 5 (𝐹 = 𝐹 → (Fun 𝐹 ↔ Fun 𝐹))
1514adantl 481 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → (Fun 𝐹 ↔ Fun 𝐹))
1613, 15mpbird 257 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → Fun 𝐹)
17 df-f1 6566 . . 3 (𝐹:𝐴1-1𝐴 ↔ (𝐹:𝐴𝐴 ∧ Fun 𝐹))
1812, 16, 17sylanbrc 583 . 2 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1𝐴)
19 simpl 482 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹 Fn 𝐴)
20 df-fo 6567 . . 3 (𝐹:𝐴onto𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴))
2119, 10, 20sylanbrc 583 . 2 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴onto𝐴)
22 df-f1o 6568 . 2 (𝐹:𝐴1-1-onto𝐴 ↔ (𝐹:𝐴1-1𝐴𝐹:𝐴onto𝐴))
2318, 21, 22sylanbrc 583 1 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  ccnv 5684  dom cdm 5685  ran crn 5686  Fun wfun 6555   Fn wfn 6556  wf 6557  1-1wf1 6558  ontowfo 6559  1-1-ontowf1o 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568
This theorem is referenced by:  mirf1o  28677  lmif1o  28803  nvocnvb  43435  dssmapf1od  44034
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