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Theorem nvof1o 7033
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 6452 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fdmrn 6537 . . . . . 6 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
31, 2sylib 219 . . . . 5 (𝐹 Fn 𝐴𝐹:dom 𝐹⟶ran 𝐹)
43adantr 481 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:dom 𝐹⟶ran 𝐹)
5 fndm 6454 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65adantr 481 . . . . 5 ((𝐹 Fn 𝐴𝐹 = 𝐹) → dom 𝐹 = 𝐴)
7 df-rn 5565 . . . . . . 7 ran 𝐹 = dom 𝐹
8 dmeq 5771 . . . . . . 7 (𝐹 = 𝐹 → dom 𝐹 = dom 𝐹)
97, 8syl5eq 2873 . . . . . 6 (𝐹 = 𝐹 → ran 𝐹 = dom 𝐹)
109, 5sylan9eqr 2883 . . . . 5 ((𝐹 Fn 𝐴𝐹 = 𝐹) → ran 𝐹 = 𝐴)
116, 10feq23d 6508 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → (𝐹:dom 𝐹⟶ran 𝐹𝐹:𝐴𝐴))
124, 11mpbid 233 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴𝐴)
131adantr 481 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → Fun 𝐹)
14 funeq 6374 . . . . 5 (𝐹 = 𝐹 → (Fun 𝐹 ↔ Fun 𝐹))
1514adantl 482 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → (Fun 𝐹 ↔ Fun 𝐹))
1613, 15mpbird 258 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → Fun 𝐹)
17 df-f1 6359 . . 3 (𝐹:𝐴1-1𝐴 ↔ (𝐹:𝐴𝐴 ∧ Fun 𝐹))
1812, 16, 17sylanbrc 583 . 2 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1𝐴)
19 simpl 483 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹 Fn 𝐴)
20 df-fo 6360 . . 3 (𝐹:𝐴onto𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴))
2119, 10, 20sylanbrc 583 . 2 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴onto𝐴)
22 df-f1o 6361 . 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 207  wa 396   = wceq 1530  ccnv 5553  dom cdm 5554  ran crn 5555  Fun wfun 6348   Fn wfn 6349  wf 6350  1-1wf1 6351  ontowfo 6352  1-1-ontowf1o 6353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361
This theorem is referenced by:  mirf1o  26388  lmif1o  26514  dssmapf1od  40251
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