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Theorem dlwwlknondlwlknonf1o 28146
 Description: 𝐹 is a bijection between the two representations of double loops of a fixed positive length on a fixed vertex. (Contributed by AV, 30-May-2022.) (Revised by AV, 1-Nov-2022.)
Hypotheses
Ref Expression
dlwwlknondlwlknonbij.v 𝑉 = (Vtx‘𝐺)
dlwwlknondlwlknonbij.w 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋)}
dlwwlknondlwlknonbij.d 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}
dlwwlknondlwlknonf1o.f 𝐹 = (𝑐𝑊 ↦ ((2nd𝑐) prefix (♯‘(1st𝑐))))
Assertion
Ref Expression
dlwwlknondlwlknonf1o ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2)) → 𝐹:𝑊1-1-onto𝐷)
Distinct variable groups:   𝐺,𝑐,𝑤   𝑁,𝑐,𝑤   𝑉,𝑐   𝑊,𝑐   𝑋,𝑐,𝑤
Allowed substitution hints:   𝐷(𝑤,𝑐)   𝐹(𝑤,𝑐)   𝑉(𝑤)   𝑊(𝑤)

Proof of Theorem dlwwlknondlwlknonf1o
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dlwwlknondlwlknonbij.w . . . 4 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋)}
2 df-3an 1086 . . . . 5 (((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋) ↔ (((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋) ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋))
32rabbii 3459 . . . 4 {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ (((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋) ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋)}
41, 3eqtri 2847 . . 3 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋) ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋)}
5 eqid 2824 . . 3 {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)}
6 dlwwlknondlwlknonf1o.f . . 3 𝐹 = (𝑐𝑊 ↦ ((2nd𝑐) prefix (♯‘(1st𝑐))))
7 eqid 2824 . . 3 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ↦ ((2nd𝑐) prefix (♯‘(1st𝑐)))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ↦ ((2nd𝑐) prefix (♯‘(1st𝑐))))
8 eluz2nn 12277 . . . 4 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℕ)
9 dlwwlknondlwlknonbij.v . . . . 5 𝑉 = (Vtx‘𝐺)
109, 5, 7clwwlknonclwlknonf1o 28143 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ↦ ((2nd𝑐) prefix (♯‘(1st𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
118, 10syl3an3 1162 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ↦ ((2nd𝑐) prefix (♯‘(1st𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
12 fveq1 6657 . . . . . . 7 (𝑦 = ((2nd𝑐) prefix (♯‘(1st𝑐))) → (𝑦‘(𝑁 − 2)) = (((2nd𝑐) prefix (♯‘(1st𝑐)))‘(𝑁 − 2)))
13123ad2ant3 1132 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ∧ 𝑦 = ((2nd𝑐) prefix (♯‘(1st𝑐)))) → (𝑦‘(𝑁 − 2)) = (((2nd𝑐) prefix (♯‘(1st𝑐)))‘(𝑁 − 2)))
14 2fveq3 6663 . . . . . . . . . . . . 13 (𝑤 = 𝑐 → (♯‘(1st𝑤)) = (♯‘(1st𝑐)))
1514eqeq1d 2826 . . . . . . . . . . . 12 (𝑤 = 𝑐 → ((♯‘(1st𝑤)) = 𝑁 ↔ (♯‘(1st𝑐)) = 𝑁))
16 fveq2 6658 . . . . . . . . . . . . . 14 (𝑤 = 𝑐 → (2nd𝑤) = (2nd𝑐))
1716fveq1d 6660 . . . . . . . . . . . . 13 (𝑤 = 𝑐 → ((2nd𝑤)‘0) = ((2nd𝑐)‘0))
1817eqeq1d 2826 . . . . . . . . . . . 12 (𝑤 = 𝑐 → (((2nd𝑤)‘0) = 𝑋 ↔ ((2nd𝑐)‘0) = 𝑋))
1915, 18anbi12d 633 . . . . . . . . . . 11 (𝑤 = 𝑐 → (((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘(1st𝑐)) = 𝑁 ∧ ((2nd𝑐)‘0) = 𝑋)))
2019elrab 3666 . . . . . . . . . 10 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑐)) = 𝑁 ∧ ((2nd𝑐)‘0) = 𝑋)))
21 simplrl 776 . . . . . . . . . . . 12 (((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑐)) = 𝑁 ∧ ((2nd𝑐)‘0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2))) → (♯‘(1st𝑐)) = 𝑁)
22 simpll 766 . . . . . . . . . . . 12 (((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑐)) = 𝑁 ∧ ((2nd𝑐)‘0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2))) → 𝑐 ∈ (ClWalks‘𝐺))
23 simpr3 1193 . . . . . . . . . . . 12 (((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑐)) = 𝑁 ∧ ((2nd𝑐)‘0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2))) → 𝑁 ∈ (ℤ‘2))
2421, 22, 233jca 1125 . . . . . . . . . . 11 (((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑐)) = 𝑁 ∧ ((2nd𝑐)‘0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2))) → ((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)))
2524ex 416 . . . . . . . . . 10 ((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st𝑐)) = 𝑁 ∧ ((2nd𝑐)‘0) = 𝑋)) → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2)) → ((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2))))
2620, 25sylbi 220 . . . . . . . . 9 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2)) → ((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2))))
2726impcom 411 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)}) → ((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)))
28 dlwwlknondlwlknonf1olem1 28145 . . . . . . . 8 (((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (((2nd𝑐) prefix (♯‘(1st𝑐)))‘(𝑁 − 2)) = ((2nd𝑐)‘(𝑁 − 2)))
2927, 28syl 17 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)}) → (((2nd𝑐) prefix (♯‘(1st𝑐)))‘(𝑁 − 2)) = ((2nd𝑐)‘(𝑁 − 2)))
30293adant3 1129 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ∧ 𝑦 = ((2nd𝑐) prefix (♯‘(1st𝑐)))) → (((2nd𝑐) prefix (♯‘(1st𝑐)))‘(𝑁 − 2)) = ((2nd𝑐)‘(𝑁 − 2)))
3113, 30eqtrd 2859 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ∧ 𝑦 = ((2nd𝑐) prefix (♯‘(1st𝑐)))) → (𝑦‘(𝑁 − 2)) = ((2nd𝑐)‘(𝑁 − 2)))
3231eqeq1d 2826 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ∧ 𝑦 = ((2nd𝑐) prefix (♯‘(1st𝑐)))) → ((𝑦‘(𝑁 − 2)) = 𝑋 ↔ ((2nd𝑐)‘(𝑁 − 2)) = 𝑋))
33 nfv 1916 . . . . 5 𝑤((2nd𝑐)‘(𝑁 − 2)) = 𝑋
3416fveq1d 6660 . . . . . 6 (𝑤 = 𝑐 → ((2nd𝑤)‘(𝑁 − 2)) = ((2nd𝑐)‘(𝑁 − 2)))
3534eqeq1d 2826 . . . . 5 (𝑤 = 𝑐 → (((2nd𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2nd𝑐)‘(𝑁 − 2)) = 𝑋))
3633, 35sbiev 2332 . . . 4 ([𝑐 / 𝑤]((2nd𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2nd𝑐)‘(𝑁 − 2)) = 𝑋)
3732, 36syl6bbr 292 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ∧ 𝑦 = ((2nd𝑐) prefix (♯‘(1st𝑐)))) → ((𝑦‘(𝑁 − 2)) = 𝑋 ↔ [𝑐 / 𝑤]((2nd𝑤)‘(𝑁 − 2)) = 𝑋))
384, 5, 6, 7, 11, 37f1ossf1o 6878 . 2 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2)) → 𝐹:𝑊1-1-onto→{𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋})
39 dlwwlknondlwlknonbij.d . . . 4 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}
40 fveq1 6657 . . . . . 6 (𝑤 = 𝑦 → (𝑤‘(𝑁 − 2)) = (𝑦‘(𝑁 − 2)))
4140eqeq1d 2826 . . . . 5 (𝑤 = 𝑦 → ((𝑤‘(𝑁 − 2)) = 𝑋 ↔ (𝑦‘(𝑁 − 2)) = 𝑋))
4241cbvrabv 3477 . . . 4 {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} = {𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋}
4339, 42eqtri 2847 . . 3 𝐷 = {𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋}
44 f1oeq3 6594 . . 3 (𝐷 = {𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋} → (𝐹:𝑊1-1-onto𝐷𝐹:𝑊1-1-onto→{𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋}))
4543, 44ax-mp 5 . 2 (𝐹:𝑊1-1-onto𝐷𝐹:𝑊1-1-onto→{𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋})
4638, 45sylibr 237 1 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2)) → 𝐹:𝑊1-1-onto𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  [wsb 2070   ∈ wcel 2115  {crab 3137   ↦ cmpt 5132  –1-1-onto→wf1o 6342  ‘cfv 6343  (class class class)co 7145  1st c1st 7677  2nd c2nd 7678  0cc0 10529   − cmin 10862  ℕcn 11630  2c2 11685  ℤ≥cuz 12236  ♯chash 13691   prefix cpfx 14028  Vtxcvtx 26785  USPGraphcuspgr 26937  ClWalkscclwlks 27555  ClWWalksNOncclwwlknon 27868 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7571  df-1st 7679  df-2nd 7680  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-1o 8092  df-2o 8093  df-oadd 8096  df-er 8279  df-map 8398  df-pm 8399  df-en 8500  df-dom 8501  df-sdom 8502  df-fin 8503  df-dju 9321  df-card 9359  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11693  df-n0 11891  df-xnn0 11961  df-z 11975  df-uz 12237  df-rp 12383  df-fz 12891  df-fzo 13034  df-hash 13692  df-word 13863  df-lsw 13911  df-concat 13919  df-s1 13946  df-substr 13999  df-pfx 14029  df-edg 26837  df-uhgr 26847  df-upgr 26871  df-uspgr 26939  df-wlks 27385  df-clwlks 27556  df-clwwlk 27763  df-clwwlkn 27806  df-clwwlknon 27869 This theorem is referenced by:  dlwwlknondlwlknonen  28147
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