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Theorem clwwlknonclwlknonf1o 27547
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.)
Hypotheses
Ref Expression
clwwlknonclwlknonf1o.v 𝑉 = (Vtx‘𝐺)
clwwlknonclwlknonf1o.w 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)}
clwwlknonclwlknonf1o.f 𝐹 = (𝑐𝑊 ↦ ((2nd𝑐) substr ⟨0, (♯‘(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   𝑊,𝑐,𝑤   𝑋,𝑐,𝑤
Allowed substitution hints:   𝐹(𝑤,𝑐)

Proof of Theorem clwwlknonclwlknonf1o
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 eqid 2770 . . . 4 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩))
2 eqid 2770 . . . . . 6 (1st𝑐) = (1st𝑐)
3 eqid 2770 . . . . . 6 (2nd𝑐) = (2nd𝑐)
4 eqid 2770 . . . . . 6 {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}
52, 3, 4, 1clwlknf1oclwwlkn 27252 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)):{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}–1-1-onto→(𝑁 ClWWalksN 𝐺))
653adant2 1124 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)):{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}–1-1-onto→(𝑁 ClWWalksN 𝐺))
7 fveq1 6331 . . . . . . 7 (𝑠 = ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩) → (𝑠‘0) = (((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)‘0))
873ad2ant3 1128 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∧ 𝑠 = ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) → (𝑠‘0) = (((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)‘0))
9 fveq2 6332 . . . . . . . . . . . . 13 (𝑤 = 𝑐 → (1st𝑤) = (1st𝑐))
109fveq2d 6336 . . . . . . . . . . . 12 (𝑤 = 𝑐 → (♯‘(1st𝑤)) = (♯‘(1st𝑐)))
1110eqeq1d 2772 . . . . . . . . . . 11 (𝑤 = 𝑐 → ((♯‘(1st𝑤)) = 𝑁 ↔ (♯‘(1st𝑐)) = 𝑁))
1211elrab 3513 . . . . . . . . . 10 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (♯‘(1st𝑐)) = 𝑁))
13 clwlkwlk 26905 . . . . . . . . . . . 12 (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
14 wlkcpr 26758 . . . . . . . . . . . . 13 (𝑐 ∈ (Walks‘𝐺) ↔ (1st𝑐)(Walks‘𝐺)(2nd𝑐))
15 eqid 2770 . . . . . . . . . . . . . . . . 17 (Vtx‘𝐺) = (Vtx‘𝐺)
1615wlkpwrd 26747 . . . . . . . . . . . . . . . 16 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
17163ad2ant1 1126 . . . . . . . . . . . . . . 15 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → (2nd𝑐) ∈ Word (Vtx‘𝐺))
18 elnnuz 11925 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ‘1))
19 eluzfz2 12555 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
2018, 19sylbi 207 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ → 𝑁 ∈ (1...𝑁))
21 fzelp1 12599 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (1...𝑁) → 𝑁 ∈ (1...(𝑁 + 1)))
2220, 21syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1)))
23223ad2ant3 1128 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝑁 ∈ (1...(𝑁 + 1)))
24233ad2ant3 1128 . . . . . . . . . . . . . . . . 17 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → 𝑁 ∈ (1...(𝑁 + 1)))
25 id 22 . . . . . . . . . . . . . . . . . . 19 ((♯‘(1st𝑐)) = 𝑁 → (♯‘(1st𝑐)) = 𝑁)
26 oveq1 6799 . . . . . . . . . . . . . . . . . . . 20 ((♯‘(1st𝑐)) = 𝑁 → ((♯‘(1st𝑐)) + 1) = (𝑁 + 1))
2726oveq2d 6808 . . . . . . . . . . . . . . . . . . 19 ((♯‘(1st𝑐)) = 𝑁 → (1...((♯‘(1st𝑐)) + 1)) = (1...(𝑁 + 1)))
2825, 27eleq12d 2843 . . . . . . . . . . . . . . . . . 18 ((♯‘(1st𝑐)) = 𝑁 → ((♯‘(1st𝑐)) ∈ (1...((♯‘(1st𝑐)) + 1)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
29283ad2ant2 1127 . . . . . . . . . . . . . . . . 17 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → ((♯‘(1st𝑐)) ∈ (1...((♯‘(1st𝑐)) + 1)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3024, 29mpbird 247 . . . . . . . . . . . . . . . 16 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → (♯‘(1st𝑐)) ∈ (1...((♯‘(1st𝑐)) + 1)))
31 wlklenvp1 26748 . . . . . . . . . . . . . . . . . . 19 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (♯‘(2nd𝑐)) = ((♯‘(1st𝑐)) + 1))
3231oveq2d 6808 . . . . . . . . . . . . . . . . . 18 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (1...(♯‘(2nd𝑐))) = (1...((♯‘(1st𝑐)) + 1)))
3332eleq2d 2835 . . . . . . . . . . . . . . . . 17 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → ((♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐))) ↔ (♯‘(1st𝑐)) ∈ (1...((♯‘(1st𝑐)) + 1))))
34333ad2ant1 1126 . . . . . . . . . . . . . . . 16 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → ((♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐))) ↔ (♯‘(1st𝑐)) ∈ (1...((♯‘(1st𝑐)) + 1))))
3530, 34mpbird 247 . . . . . . . . . . . . . . 15 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐))))
3617, 35jca 495 . . . . . . . . . . . . . 14 (((1st𝑐)(Walks‘𝐺)(2nd𝑐) ∧ (♯‘(1st𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ)) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))))
37363exp 1111 . . . . . . . . . . . . 13 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → ((♯‘(1st𝑐)) = 𝑁 → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))))))
3814, 37sylbi 207 . . . . . . . . . . . 12 (𝑐 ∈ (Walks‘𝐺) → ((♯‘(1st𝑐)) = 𝑁 → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))))))
3913, 38syl 17 . . . . . . . . . . 11 (𝑐 ∈ (ClWalks‘𝐺) → ((♯‘(1st𝑐)) = 𝑁 → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))))))
4039imp 393 . . . . . . . . . 10 ((𝑐 ∈ (ClWalks‘𝐺) ∧ (♯‘(1st𝑐)) = 𝑁) → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐))))))
4112, 40sylbi 207 . . . . . . . . 9 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} → ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐))))))
4241impcom 394 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}) → ((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))))
43 swrd0fv0 13648 . . . . . . . 8 (((2nd𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st𝑐)) ∈ (1...(♯‘(2nd𝑐)))) → (((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)‘0) = ((2nd𝑐)‘0))
4442, 43syl 17 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}) → (((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)‘0) = ((2nd𝑐)‘0))
45443adant3 1125 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∧ 𝑠 = ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) → (((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)‘0) = ((2nd𝑐)‘0))
468, 45eqtrd 2804 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∧ 𝑠 = ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) → (𝑠‘0) = ((2nd𝑐)‘0))
4746eqeq1d 2772 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∧ 𝑠 = ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) → ((𝑠‘0) = 𝑋 ↔ ((2nd𝑐)‘0) = 𝑋))
481, 6, 47f1oresrab 6537 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) ↾ {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑐)‘0) = 𝑋}):{𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑐)‘0) = 𝑋}–1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋})
49 nfrab1 3270 . . . . . . 7 𝑤{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}
50 nfcv 2912 . . . . . . 7 𝑐{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}
51 nfv 1994 . . . . . . 7 𝑐((2nd𝑤)‘0) = 𝑋
52 nfv 1994 . . . . . . 7 𝑤((2nd𝑐)‘0) = 𝑋
53 fveq2 6332 . . . . . . . . 9 (𝑤 = 𝑐 → (2nd𝑤) = (2nd𝑐))
5453fveq1d 6334 . . . . . . . 8 (𝑤 = 𝑐 → ((2nd𝑤)‘0) = ((2nd𝑐)‘0))
5554eqeq1d 2772 . . . . . . 7 (𝑤 = 𝑐 → (((2nd𝑤)‘0) = 𝑋 ↔ ((2nd𝑐)‘0) = 𝑋))
5649, 50, 51, 52, 55cbvrab 3347 . . . . . 6 {𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋} = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑐)‘0) = 𝑋}
5756a1i 11 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → {𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋} = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑐)‘0) = 𝑋})
5857reseq2d 5534 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) ↾ {𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋}) = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) ↾ {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑐)‘0) = 𝑋}))
59 eqidd 2771 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋} = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋})
6058, 57, 59f1oeq123d 6274 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) ↾ {𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋}):{𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋}–1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋} ↔ ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) ↾ {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑐)‘0) = 𝑋}):{𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑐)‘0) = 𝑋}–1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋}))
6148, 60mpbird 247 . 2 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) ↾ {𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋}):{𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋}–1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋})
62 clwwlknonclwlknonf1o.v . . . . . 6 𝑉 = (Vtx‘𝐺)
63 clwwlknonclwlknonf1o.w . . . . . 6 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)}
64 clwwlknonclwlknonf1o.f . . . . . 6 𝐹 = (𝑐𝑊 ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩))
6562, 63, 64clwwlknonclwlknonf1olem 27546 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) ↾ 𝑊))
66 rabrab 3264 . . . . . . 7 {𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)}
6766, 63eqtr4i 2795 . . . . . 6 {𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋} = 𝑊
6867reseq2i 5531 . . . . 5 ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) ↾ {𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋}) = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) ↾ 𝑊)
6965, 68syl6eqr 2822 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) ↾ {𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋}))
70693adant2 1124 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) ↾ {𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋}))
7163, 66eqtr4i 2795 . . . 4 𝑊 = {𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋}
7271a1i 11 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝑊 = {𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋})
73 clwwlknon 27259 . . . 4 (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋}
7473a1i 11 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋})
7570, 72, 74f1oeq123d 6274 . 2 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝐹:𝑊1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)) ↾ {𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋}):{𝑤 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∣ ((2nd𝑤)‘0) = 𝑋}–1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋}))
7661, 75mpbird 247 1 ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝐹:𝑊1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  {crab 3064  cop 4320   class class class wbr 4784  cmpt 4861  cres 5251  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  0cc0 10137  1c1 10138   + caddc 10140  cn 11221  cuz 11887  ...cfz 12532  chash 13320  Word cword 13486   substr csubstr 13490  Vtxcvtx 26094  USPGraphcuspgr 26264  Walkscwlks 26726  ClWalkscclwlks 26900   ClWWalksN cclwwlkn 27171  ClWWalksNOncclwwlknon 27256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-ifp 1049  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-card 8964  df-cda 9191  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-n0 11494  df-xnn0 11565  df-z 11579  df-uz 11888  df-rp 12035  df-fz 12533  df-fzo 12673  df-hash 13321  df-word 13494  df-lsw 13495  df-concat 13496  df-s1 13497  df-substr 13498  df-edg 26160  df-uhgr 26173  df-upgr 26197  df-uspgr 26266  df-wlks 26729  df-clwlks 26901  df-clwwlk 27129  df-clwwlkn 27173  df-clwwlknon 27257
This theorem is referenced by:  clwwlknonclwlknonen  27548  dlwwlknondlwlknonf1o  27551
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