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Theorem nvof1o 6491
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 5948 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fdmrn 6023 . . . . . 6 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
31, 2sylib 208 . . . . 5 (𝐹 Fn 𝐴𝐹:dom 𝐹⟶ran 𝐹)
43adantr 481 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:dom 𝐹⟶ran 𝐹)
5 fndm 5950 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65adantr 481 . . . . 5 ((𝐹 Fn 𝐴𝐹 = 𝐹) → dom 𝐹 = 𝐴)
7 df-rn 5090 . . . . . . 7 ran 𝐹 = dom 𝐹
8 dmeq 5289 . . . . . . 7 (𝐹 = 𝐹 → dom 𝐹 = dom 𝐹)
97, 8syl5eq 2672 . . . . . 6 (𝐹 = 𝐹 → ran 𝐹 = dom 𝐹)
109, 5sylan9eqr 2682 . . . . 5 ((𝐹 Fn 𝐴𝐹 = 𝐹) → ran 𝐹 = 𝐴)
116, 10feq23d 5999 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → (𝐹:dom 𝐹⟶ran 𝐹𝐹:𝐴𝐴))
124, 11mpbid 222 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴𝐴)
131adantr 481 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → Fun 𝐹)
14 funeq 5869 . . . . 5 (𝐹 = 𝐹 → (Fun 𝐹 ↔ Fun 𝐹))
1514adantl 482 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → (Fun 𝐹 ↔ Fun 𝐹))
1613, 15mpbird 247 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → Fun 𝐹)
17 df-f1 5855 . . 3 (𝐹:𝐴1-1𝐴 ↔ (𝐹:𝐴𝐴 ∧ Fun 𝐹))
1812, 16, 17sylanbrc 697 . 2 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1𝐴)
19 simpl 473 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹 Fn 𝐴)
20 df-fo 5856 . . 3 (𝐹:𝐴onto𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴))
2119, 10, 20sylanbrc 697 . 2 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴onto𝐴)
22 df-f1o 5857 . 2 (𝐹:𝐴1-1-onto𝐴 ↔ (𝐹:𝐴1-1𝐴𝐹:𝐴onto𝐴))
2318, 21, 22sylanbrc 697 1 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  ccnv 5078  dom cdm 5079  ran crn 5080  Fun wfun 5844   Fn wfn 5845  wf 5846  1-1wf1 5847  ontowfo 5848  1-1-ontowf1o 5849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857
This theorem is referenced by:  mirf1o  25459  lmif1o  25582  dssmapf1od  37783
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