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Theorem nvof1o 7030
 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 6435 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fdmrn 6524 . . . . . 6 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
31, 2sylib 221 . . . . 5 (𝐹 Fn 𝐴𝐹:dom 𝐹⟶ran 𝐹)
43adantr 485 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:dom 𝐹⟶ran 𝐹)
5 fndm 6437 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65adantr 485 . . . . 5 ((𝐹 Fn 𝐴𝐹 = 𝐹) → dom 𝐹 = 𝐴)
7 df-rn 5536 . . . . . . 7 ran 𝐹 = dom 𝐹
8 dmeq 5744 . . . . . . 7 (𝐹 = 𝐹 → dom 𝐹 = dom 𝐹)
97, 8syl5eq 2806 . . . . . 6 (𝐹 = 𝐹 → ran 𝐹 = dom 𝐹)
109, 5sylan9eqr 2816 . . . . 5 ((𝐹 Fn 𝐴𝐹 = 𝐹) → ran 𝐹 = 𝐴)
116, 10feq23d 6494 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → (𝐹:dom 𝐹⟶ran 𝐹𝐹:𝐴𝐴))
124, 11mpbid 235 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴𝐴)
131adantr 485 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → Fun 𝐹)
14 funeq 6356 . . . . 5 (𝐹 = 𝐹 → (Fun 𝐹 ↔ Fun 𝐹))
1514adantl 486 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → (Fun 𝐹 ↔ Fun 𝐹))
1613, 15mpbird 260 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → Fun 𝐹)
17 df-f1 6341 . . 3 (𝐹:𝐴1-1𝐴 ↔ (𝐹:𝐴𝐴 ∧ Fun 𝐹))
1812, 16, 17sylanbrc 587 . 2 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1𝐴)
19 simpl 487 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹 Fn 𝐴)
20 df-fo 6342 . . 3 (𝐹:𝐴onto𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴))
2119, 10, 20sylanbrc 587 . 2 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴onto𝐴)
22 df-f1o 6343 . 2 (𝐹:𝐴1-1-onto𝐴 ↔ (𝐹:𝐴1-1𝐴𝐹:𝐴onto𝐴))
2318, 21, 22sylanbrc 587 1 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1-onto𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539  ◡ccnv 5524  dom cdm 5525  ran crn 5526  Fun wfun 6330   Fn wfn 6331  ⟶wf 6332  –1-1→wf1 6333  –onto→wfo 6334  –1-1-onto→wf1o 6335 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-v 3412  df-un 3864  df-in 3866  df-ss 3876  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343 This theorem is referenced by:  mirf1o  26555  lmif1o  26681  dssmapf1od  41088
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