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Theorem clwwlknonclwlknonf1o 30391
Description: 𝐹 is a bijection between the two representations of closed walks of a fixed positive length on a fixed vertex. (Contributed by AV, 26-May-2022.) (Proof shortened by AV, 7-Aug-2022.) (Revised by AV, 1-Nov-2022.)
Hypotheses
Ref Expression
clwwlknonclwlknonf1o.v 𝑉 = (Vtx‘𝐺)
clwwlknonclwlknonf1o.w 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)}
clwwlknonclwlknonf1o.f 𝐹 = (𝑐𝑊 ↦ ((2nd𝑐) prefix (♯‘(1st𝑐))))
Assertion
Ref Expression
clwwlknonclwlknonf1o ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝐹:𝑊1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
Distinct variable groups:   𝐺,𝑐,𝑤   𝑁,𝑐,𝑤   𝑉,𝑐   𝑊,𝑐   𝑋,𝑐,𝑤
Allowed substitution hints:   𝐹(𝑤,𝑐)   𝑉(𝑤)   𝑊(𝑤)

Proof of Theorem clwwlknonclwlknonf1o
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 clwwlknonclwlknonf1o.w . . 3 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)}
2 eqid 2735 . . 3 {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}
3 clwwlknonclwlknonf1o.f . . 3 𝐹 = (𝑐𝑊 ↦ ((2nd𝑐) prefix (♯‘(1st𝑐))))
4 eqid 2735 . . 3 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) prefix (♯‘(1st𝑐)))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) prefix (♯‘(1st𝑐))))
5 eqid 2735 . . . . 5 (1st𝑐) = (1st𝑐)
6 eqid 2735 . . . . 5 (2nd𝑐) = (2nd𝑐)
75, 6, 2, 4clwlknf1oclwwlkn 30113 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) prefix (♯‘(1st𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}–1-1-onto→(𝑁 ClWWalksN 𝐺))
873adant2 1130 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) prefix (♯‘(1st𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}–1-1-onto→(𝑁 ClWWalksN 𝐺))
9 fveq1 6906 . . . . . . 7 (𝑠 = ((2nd𝑐) prefix (♯‘(1st𝑐))) → (𝑠‘0) = (((2nd𝑐) prefix (♯‘(1st𝑐)))‘0))
1093ad2ant3 1134 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∧ 𝑠 = ((2nd𝑐) prefix (♯‘(1st𝑐)))) → (𝑠‘0) = (((2nd𝑐) prefix (♯‘(1st𝑐)))‘0))
11 2fveq3 6912 . . . . . . . . . . . 12 (𝑤 = 𝑐 → (♯‘(1st𝑤)) = (♯‘(1st𝑐)))
1211eqeq1d 2737 . . . . . . . . . . 11 (𝑤 = 𝑐 → ((♯‘(1st𝑤)) = 𝑁 ↔ (♯‘(1st𝑐)) = 𝑁))
1312elrab 3695 . . . . . . . . . 10 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (♯‘(1st𝑐)) = 𝑁))
14 clwlkwlk 29808 . . . . . . . . . . . 12 (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
15 wlkcpr 29662 . . . . . . . . . . . . 13 (𝑐 ∈ (Walks‘𝐺) ↔ (1st𝑐)(Walks‘𝐺)(2nd𝑐))
16 eqid 2735 . . . . . . . . . . . . . . . . 17 (Vtx‘𝐺) = (Vtx‘𝐺)
1716wlkpwrd 29650 . . . . . . . . . . . . . . . 16 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
18173ad2ant1 1132 . . . . . . . . . . . . . . 15 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
19 elnnuz 12920 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ‘1))
20 eluzfz2 13569 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
2119, 20sylbi 217 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ → 𝑁 ∈ (1...𝑁))
22 fzelp1 13613 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (1...𝑁) → 𝑁 ∈ (1...(𝑁 + 1)))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
24233ad2ant3 1134 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝑁 ∈ (1...(𝑁 + 1)))
25243ad2ant3 1134 . . . . . . . . . . . . . . . . 17 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → 𝑁 ∈ (1...(𝑁 + 1)))
26 id 22 . . . . . . . . . . . . . . . . . . 19 ((♯‘(1st𝑐)) = 𝑁 → (♯‘(1st𝑐)) = 𝑁)
27 oveq1 7438 . . . . . . . . . . . . . . . . . . . 20 ((♯‘(1st𝑐)) = 𝑁 → ((♯‘(1st𝑐)) + 1) = (𝑁 + 1))
2827oveq2d 7447 . . . . . . . . . . . . . . . . . . 19 ((♯‘(1st𝑐)) = 𝑁 → (1...((♯‘(1st𝑐)) + 1)) = (1...(𝑁 + 1)))
2926, 28eleq12d 2833 . . . . . . . . . . . . . . . . . 18 ((♯‘(1st𝑐)) = 𝑁 → ((♯‘(1st𝑐)) ∈ (1...((♯‘(1st𝑐)) + 1)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
30293ad2ant2 1133 . . . . . . . . . . . . . . . . 17 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → ((♯‘(1st𝑐)) ∈ (1...((♯‘(1st𝑐)) + 1)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3125, 30mpbird 257 . . . . . . . . . . . . . . . 16 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → (♯‘(1st𝑐)) ∈ (1...((♯‘(1st𝑐)) + 1)))
32 wlklenvp1 29651 . . . . . . . . . . . . . . . . . . 19 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (♯‘(2nd𝑐)) = ((♯‘(1st𝑐)) + 1))
3332oveq2d 7447 . . . . . . . . . . . . . . . . . 18 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (1...(♯‘(2nd𝑐))) = (1...((♯‘(1st𝑐)) + 1)))
3433eleq2d 2825 . . . . . . . . . . . . . . . . 17 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → ((♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐))) ↔ (♯‘(1st𝑐)) ∈ (1...((♯‘(1st𝑐)) + 1))))
35343ad2ant1 1132 . . . . . . . . . . . . . . . 16 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → ((♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐))) ↔ (♯‘(1st𝑐)) ∈ (1...((♯‘(1st𝑐)) + 1))))
3631, 35mpbird 257 . . . . . . . . . . . . . . 15 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐))))
3718, 36jca 511 . . . . . . . . . . . . . 14 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))))
38373exp 1118 . . . . . . . . . . . . 13 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → ((♯‘(1st𝑐)) = 𝑁 → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))))))
3915, 38sylbi 217 . . . . . . . . . . . 12 (𝑐 ∈ (Walks‘𝐺) → ((♯‘(1st𝑐)) = 𝑁 → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))))))
4014, 39syl 17 . . . . . . . . . . 11 (𝑐 ∈ (ClWalks‘𝐺) → ((♯‘(1st𝑐)) = 𝑁 → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))))))
4140imp 406 . . . . . . . . . 10 ((𝑐 ∈ (ClWalks‘𝐺) ∧ (♯‘(1st𝑐)) = 𝑁) → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐))))))
4213, 41sylbi 217 . . . . . . . . 9 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐))))))
4342impcom 407 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))))
44 pfxfv0 14727 . . . . . . . 8 (((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))) → (((2nd𝑐) prefix (♯‘(1st𝑐)))‘0) = ((2nd𝑐)‘0))
4543, 44syl 17 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}) → (((2nd𝑐) prefix (♯‘(1st𝑐)))‘0) = ((2nd𝑐)‘0))
46453adant3 1131 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∧ 𝑠 = ((2nd𝑐) prefix (♯‘(1st𝑐)))) → (((2nd𝑐) prefix (♯‘(1st𝑐)))‘0) = ((2nd𝑐)‘0))
4710, 46eqtrd 2775 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∧ 𝑠 = ((2nd𝑐) prefix (♯‘(1st𝑐)))) → (𝑠‘0) = ((2nd𝑐)‘0))
4847eqeq1d 2737 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∧ 𝑠 = ((2nd𝑐) prefix (♯‘(1st𝑐)))) → ((𝑠‘0) = 𝑋 ↔ ((2nd𝑐)‘0) = 𝑋))
49 nfv 1912 . . . . 5 𝑤((2nd𝑐)‘0) = 𝑋
50 fveq2 6907 . . . . . . 7 (𝑤 = 𝑐 → (2nd𝑤) = (2nd𝑐))
5150fveq1d 6909 . . . . . 6 (𝑤 = 𝑐 → ((2nd𝑤)‘0) = ((2nd𝑐)‘0))
5251eqeq1d 2737 . . . . 5 (𝑤 = 𝑐 → (((2nd𝑤)‘0) = 𝑋 ↔ ((2nd𝑐)‘0) = 𝑋))
5349, 52sbiev 2313 . . . 4 ([𝑐 / 𝑤]((2nd𝑤)‘0) = 𝑋 ↔ ((2nd𝑐)‘0) = 𝑋)
5448, 53bitr4di 289 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∧ 𝑠 = ((2nd𝑐) prefix (♯‘(1st𝑐)))) → ((𝑠‘0) = 𝑋 ↔ [𝑐 / 𝑤]((2nd𝑤)‘0) = 𝑋))
551, 2, 3, 4, 8, 54f1ossf1o 7148 . 2 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝐹:𝑊1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋})
56 clwwlknon 30119 . . 3 (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋}
57 f1oeq3 6839 . . 3 ((𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋} → (𝐹:𝑊1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ 𝐹:𝑊1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋}))
5856, 57ax-mp 5 . 2 (𝐹:𝑊1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ 𝐹:𝑊1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋})
5955, 58sylibr 234 1 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝐹:𝑊1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  [wsb 2062  wcel 2106  {crab 3433   class class class wbr 5148  cmpt 5231  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  0cc0 11153  1c1 11154   + caddc 11156  cn 12264  cuz 12876  ...cfz 13544  chash 14366  Word cword 14549   prefix cpfx 14705  Vtxcvtx 29028  USPGraphcuspgr 29180  Walkscwlks 29629  ClWalkscclwlks 29803   ClWWalksN cclwwlkn 30053  ClWWalksNOncclwwlknon 30116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-lsw 14598  df-concat 14606  df-s1 14631  df-substr 14676  df-pfx 14706  df-edg 29080  df-uhgr 29090  df-upgr 29114  df-uspgr 29182  df-wlks 29632  df-clwlks 29804  df-clwwlk 30011  df-clwwlkn 30054  df-clwwlknon 30117
This theorem is referenced by:  clwwlknonclwlknonen  30392  dlwwlknondlwlknonf1o  30394
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