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Theorem nvof1o 6732
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 6168 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fdmrn 6248 . . . . . 6 (Fun 𝐹𝐹:dom 𝐹⟶ran 𝐹)
31, 2sylib 209 . . . . 5 (𝐹 Fn 𝐴𝐹:dom 𝐹⟶ran 𝐹)
43adantr 472 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:dom 𝐹⟶ran 𝐹)
5 fndm 6170 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65adantr 472 . . . . 5 ((𝐹 Fn 𝐴𝐹 = 𝐹) → dom 𝐹 = 𝐴)
7 df-rn 5290 . . . . . . 7 ran 𝐹 = dom 𝐹
8 dmeq 5494 . . . . . . 7 (𝐹 = 𝐹 → dom 𝐹 = dom 𝐹)
97, 8syl5eq 2811 . . . . . 6 (𝐹 = 𝐹 → ran 𝐹 = dom 𝐹)
109, 5sylan9eqr 2821 . . . . 5 ((𝐹 Fn 𝐴𝐹 = 𝐹) → ran 𝐹 = 𝐴)
116, 10feq23d 6220 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → (𝐹:dom 𝐹⟶ran 𝐹𝐹:𝐴𝐴))
124, 11mpbid 223 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴𝐴)
131adantr 472 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → Fun 𝐹)
14 funeq 6090 . . . . 5 (𝐹 = 𝐹 → (Fun 𝐹 ↔ Fun 𝐹))
1514adantl 473 . . . 4 ((𝐹 Fn 𝐴𝐹 = 𝐹) → (Fun 𝐹 ↔ Fun 𝐹))
1613, 15mpbird 248 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → Fun 𝐹)
17 df-f1 6075 . . 3 (𝐹:𝐴1-1𝐴 ↔ (𝐹:𝐴𝐴 ∧ Fun 𝐹))
1812, 16, 17sylanbrc 578 . 2 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1𝐴)
19 simpl 474 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹 Fn 𝐴)
20 df-fo 6076 . . 3 (𝐹:𝐴onto𝐴 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐴))
2119, 10, 20sylanbrc 578 . 2 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴onto𝐴)
22 df-f1o 6077 . 2 (𝐹:𝐴1-1-onto𝐴 ↔ (𝐹:𝐴1-1𝐴𝐹:𝐴onto𝐴))
2318, 21, 22sylanbrc 578 1 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  ccnv 5278  dom cdm 5279  ran crn 5280  Fun wfun 6064   Fn wfn 6065  wf 6066  1-1wf1 6067  ontowfo 6068  1-1-ontowf1o 6069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077
This theorem is referenced by:  mirf1o  25869  lmif1o  25992  dssmapf1od  39013
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