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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-onto→wf1o 6030  ‘cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  0cc0 10137  1c1 10138   + caddc 10140  ℕcn 11221  ℤ≥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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