| Step | Hyp | Ref
| Expression |
| 1 | | clwwlknonclwlknonf1o.w |
. . 3
⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st
‘𝑤)) = 𝑁 ∧ ((2nd
‘𝑤)‘0) = 𝑋)} |
| 2 | | eqid 2737 |
. . 3
⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣
(♯‘(1st ‘𝑤)) = 𝑁} = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁} |
| 3 | | clwwlknonclwlknonf1o.f |
. . 3
⊢ 𝐹 = (𝑐 ∈ 𝑊 ↦ ((2nd ‘𝑐) prefix
(♯‘(1st ‘𝑐)))) |
| 4 | | eqid 2737 |
. . 3
⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁} ↦ ((2nd
‘𝑐) prefix
(♯‘(1st ‘𝑐)))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁} ↦ ((2nd
‘𝑐) prefix
(♯‘(1st ‘𝑐)))) |
| 5 | | eqid 2737 |
. . . . 5
⊢
(1st ‘𝑐) = (1st ‘𝑐) |
| 6 | | eqid 2737 |
. . . . 5
⊢
(2nd ‘𝑐) = (2nd ‘𝑐) |
| 7 | 5, 6, 2, 4 | clwlknf1oclwwlkn 30103 |
. . . 4
⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁} ↦ ((2nd
‘𝑐) prefix
(♯‘(1st ‘𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁}–1-1-onto→(𝑁 ClWWalksN 𝐺)) |
| 8 | 7 | 3adant2 1132 |
. . 3
⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁} ↦ ((2nd
‘𝑐) prefix
(♯‘(1st ‘𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁}–1-1-onto→(𝑁 ClWWalksN 𝐺)) |
| 9 | | fveq1 6905 |
. . . . . . 7
⊢ (𝑠 = ((2nd ‘𝑐) prefix
(♯‘(1st ‘𝑐))) → (𝑠‘0) = (((2nd ‘𝑐) prefix
(♯‘(1st ‘𝑐)))‘0)) |
| 10 | 9 | 3ad2ant3 1136 |
. . . . . 6
⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁} ∧ 𝑠 = ((2nd ‘𝑐) prefix (♯‘(1st
‘𝑐)))) → (𝑠‘0) = (((2nd
‘𝑐) prefix
(♯‘(1st ‘𝑐)))‘0)) |
| 11 | | 2fveq3 6911 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑐 → (♯‘(1st
‘𝑤)) =
(♯‘(1st ‘𝑐))) |
| 12 | 11 | eqeq1d 2739 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑐 → ((♯‘(1st
‘𝑤)) = 𝑁 ↔
(♯‘(1st ‘𝑐)) = 𝑁)) |
| 13 | 12 | elrab 3692 |
. . . . . . . . . 10
⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (♯‘(1st
‘𝑐)) = 𝑁)) |
| 14 | | clwlkwlk 29795 |
. . . . . . . . . . . 12
⊢ (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺)) |
| 15 | | wlkcpr 29647 |
. . . . . . . . . . . . 13
⊢ (𝑐 ∈ (Walks‘𝐺) ↔ (1st
‘𝑐)(Walks‘𝐺)(2nd ‘𝑐)) |
| 16 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
⊢
(Vtx‘𝐺) =
(Vtx‘𝐺) |
| 17 | 16 | wlkpwrd 29635 |
. . . . . . . . . . . . . . . 16
⊢
((1st ‘𝑐)(Walks‘𝐺)(2nd ‘𝑐) → (2nd ‘𝑐) ∈ Word (Vtx‘𝐺)) |
| 18 | 17 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘𝑐)(Walks‘𝐺)(2nd ‘𝑐) ∧ (♯‘(1st
‘𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (2nd
‘𝑐) ∈ Word
(Vtx‘𝐺)) |
| 19 | | elnnuz 12922 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈
(ℤ≥‘1)) |
| 20 | | eluzfz2 13572 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑁 ∈
(ℤ≥‘1) → 𝑁 ∈ (1...𝑁)) |
| 21 | 19, 20 | sylbi 217 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...𝑁)) |
| 22 | | fzelp1 13616 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ (1...𝑁) → 𝑁 ∈ (1...(𝑁 + 1))) |
| 23 | 21, 22 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (1...(𝑁 + 1))) |
| 24 | 23 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ (1...(𝑁 + 1))) |
| 25 | 24 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . . . 17
⊢
(((1st ‘𝑐)(Walks‘𝐺)(2nd ‘𝑐) ∧ (♯‘(1st
‘𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → 𝑁 ∈ (1...(𝑁 + 1))) |
| 26 | | id 22 |
. . . . . . . . . . . . . . . . . . 19
⊢
((♯‘(1st ‘𝑐)) = 𝑁 → (♯‘(1st
‘𝑐)) = 𝑁) |
| 27 | | oveq1 7438 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((♯‘(1st ‘𝑐)) = 𝑁 → ((♯‘(1st
‘𝑐)) + 1) = (𝑁 + 1)) |
| 28 | 27 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . 19
⊢
((♯‘(1st ‘𝑐)) = 𝑁 → (1...((♯‘(1st
‘𝑐)) + 1)) =
(1...(𝑁 +
1))) |
| 29 | 26, 28 | eleq12d 2835 |
. . . . . . . . . . . . . . . . . 18
⊢
((♯‘(1st ‘𝑐)) = 𝑁 → ((♯‘(1st
‘𝑐)) ∈
(1...((♯‘(1st ‘𝑐)) + 1)) ↔ 𝑁 ∈ (1...(𝑁 + 1)))) |
| 30 | 29 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . . 17
⊢
(((1st ‘𝑐)(Walks‘𝐺)(2nd ‘𝑐) ∧ (♯‘(1st
‘𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) →
((♯‘(1st ‘𝑐)) ∈
(1...((♯‘(1st ‘𝑐)) + 1)) ↔ 𝑁 ∈ (1...(𝑁 + 1)))) |
| 31 | 25, 30 | mpbird 257 |
. . . . . . . . . . . . . . . 16
⊢
(((1st ‘𝑐)(Walks‘𝐺)(2nd ‘𝑐) ∧ (♯‘(1st
‘𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) →
(♯‘(1st ‘𝑐)) ∈
(1...((♯‘(1st ‘𝑐)) + 1))) |
| 32 | | wlklenvp1 29636 |
. . . . . . . . . . . . . . . . . . 19
⊢
((1st ‘𝑐)(Walks‘𝐺)(2nd ‘𝑐) → (♯‘(2nd
‘𝑐)) =
((♯‘(1st ‘𝑐)) + 1)) |
| 33 | 32 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . 18
⊢
((1st ‘𝑐)(Walks‘𝐺)(2nd ‘𝑐) → (1...(♯‘(2nd
‘𝑐))) =
(1...((♯‘(1st ‘𝑐)) + 1))) |
| 34 | 33 | eleq2d 2827 |
. . . . . . . . . . . . . . . . 17
⊢
((1st ‘𝑐)(Walks‘𝐺)(2nd ‘𝑐) → ((♯‘(1st
‘𝑐)) ∈
(1...(♯‘(2nd ‘𝑐))) ↔ (♯‘(1st
‘𝑐)) ∈
(1...((♯‘(1st ‘𝑐)) + 1)))) |
| 35 | 34 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . . 16
⊢
(((1st ‘𝑐)(Walks‘𝐺)(2nd ‘𝑐) ∧ (♯‘(1st
‘𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) →
((♯‘(1st ‘𝑐)) ∈ (1...(♯‘(2nd
‘𝑐))) ↔
(♯‘(1st ‘𝑐)) ∈
(1...((♯‘(1st ‘𝑐)) + 1)))) |
| 36 | 31, 35 | mpbird 257 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘𝑐)(Walks‘𝐺)(2nd ‘𝑐) ∧ (♯‘(1st
‘𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) →
(♯‘(1st ‘𝑐)) ∈ (1...(♯‘(2nd
‘𝑐)))) |
| 37 | 18, 36 | jca 511 |
. . . . . . . . . . . . . 14
⊢
(((1st ‘𝑐)(Walks‘𝐺)(2nd ‘𝑐) ∧ (♯‘(1st
‘𝑐)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → ((2nd
‘𝑐) ∈ Word
(Vtx‘𝐺) ∧
(♯‘(1st ‘𝑐)) ∈ (1...(♯‘(2nd
‘𝑐))))) |
| 38 | 37 | 3exp 1120 |
. . . . . . . . . . . . 13
⊢
((1st ‘𝑐)(Walks‘𝐺)(2nd ‘𝑐) → ((♯‘(1st
‘𝑐)) = 𝑁 → ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((2nd
‘𝑐) ∈ Word
(Vtx‘𝐺) ∧
(♯‘(1st ‘𝑐)) ∈ (1...(♯‘(2nd
‘𝑐))))))) |
| 39 | 15, 38 | sylbi 217 |
. . . . . . . . . . . 12
⊢ (𝑐 ∈ (Walks‘𝐺) →
((♯‘(1st ‘𝑐)) = 𝑁 → ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((2nd
‘𝑐) ∈ Word
(Vtx‘𝐺) ∧
(♯‘(1st ‘𝑐)) ∈ (1...(♯‘(2nd
‘𝑐))))))) |
| 40 | 14, 39 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑐 ∈ (ClWalks‘𝐺) →
((♯‘(1st ‘𝑐)) = 𝑁 → ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((2nd
‘𝑐) ∈ Word
(Vtx‘𝐺) ∧
(♯‘(1st ‘𝑐)) ∈ (1...(♯‘(2nd
‘𝑐))))))) |
| 41 | 40 | imp 406 |
. . . . . . . . . 10
⊢ ((𝑐 ∈ (ClWalks‘𝐺) ∧
(♯‘(1st ‘𝑐)) = 𝑁) → ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((2nd
‘𝑐) ∈ Word
(Vtx‘𝐺) ∧
(♯‘(1st ‘𝑐)) ∈ (1...(♯‘(2nd
‘𝑐)))))) |
| 42 | 13, 41 | sylbi 217 |
. . . . . . . . 9
⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁} → ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ((2nd
‘𝑐) ∈ Word
(Vtx‘𝐺) ∧
(♯‘(1st ‘𝑐)) ∈ (1...(♯‘(2nd
‘𝑐)))))) |
| 43 | 42 | impcom 407 |
. . . . . . . 8
⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁}) → ((2nd
‘𝑐) ∈ Word
(Vtx‘𝐺) ∧
(♯‘(1st ‘𝑐)) ∈ (1...(♯‘(2nd
‘𝑐))))) |
| 44 | | pfxfv0 14730 |
. . . . . . . 8
⊢
(((2nd ‘𝑐) ∈ Word (Vtx‘𝐺) ∧ (♯‘(1st
‘𝑐)) ∈
(1...(♯‘(2nd ‘𝑐)))) → (((2nd ‘𝑐) prefix
(♯‘(1st ‘𝑐)))‘0) = ((2nd ‘𝑐)‘0)) |
| 45 | 43, 44 | syl 17 |
. . . . . . 7
⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁}) → (((2nd
‘𝑐) prefix
(♯‘(1st ‘𝑐)))‘0) = ((2nd ‘𝑐)‘0)) |
| 46 | 45 | 3adant3 1133 |
. . . . . 6
⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁} ∧ 𝑠 = ((2nd ‘𝑐) prefix (♯‘(1st
‘𝑐)))) →
(((2nd ‘𝑐)
prefix (♯‘(1st ‘𝑐)))‘0) = ((2nd ‘𝑐)‘0)) |
| 47 | 10, 46 | eqtrd 2777 |
. . . . 5
⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁} ∧ 𝑠 = ((2nd ‘𝑐) prefix (♯‘(1st
‘𝑐)))) → (𝑠‘0) = ((2nd
‘𝑐)‘0)) |
| 48 | 47 | eqeq1d 2739 |
. . . 4
⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁} ∧ 𝑠 = ((2nd ‘𝑐) prefix (♯‘(1st
‘𝑐)))) → ((𝑠‘0) = 𝑋 ↔ ((2nd ‘𝑐)‘0) = 𝑋)) |
| 49 | | nfv 1914 |
. . . . 5
⊢
Ⅎ𝑤((2nd ‘𝑐)‘0) = 𝑋 |
| 50 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑤 = 𝑐 → (2nd ‘𝑤) = (2nd ‘𝑐)) |
| 51 | 50 | fveq1d 6908 |
. . . . . 6
⊢ (𝑤 = 𝑐 → ((2nd ‘𝑤)‘0) = ((2nd
‘𝑐)‘0)) |
| 52 | 51 | eqeq1d 2739 |
. . . . 5
⊢ (𝑤 = 𝑐 → (((2nd ‘𝑤)‘0) = 𝑋 ↔ ((2nd ‘𝑐)‘0) = 𝑋)) |
| 53 | 49, 52 | sbiev 2314 |
. . . 4
⊢ ([𝑐 / 𝑤]((2nd ‘𝑤)‘0) = 𝑋 ↔ ((2nd ‘𝑐)‘0) = 𝑋) |
| 54 | 48, 53 | bitr4di 289 |
. . 3
⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st
‘𝑤)) = 𝑁} ∧ 𝑠 = ((2nd ‘𝑐) prefix (♯‘(1st
‘𝑐)))) → ((𝑠‘0) = 𝑋 ↔ [𝑐 / 𝑤]((2nd ‘𝑤)‘0) = 𝑋)) |
| 55 | 1, 2, 3, 4, 8, 54 | f1ossf1o 7148 |
. 2
⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝐹:𝑊–1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋}) |
| 56 | | clwwlknon 30109 |
. . 3
⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋} |
| 57 | | f1oeq3 6838 |
. . 3
⊢ ((𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋} → (𝐹:𝑊–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ 𝐹:𝑊–1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋})) |
| 58 | 56, 57 | ax-mp 5 |
. 2
⊢ (𝐹:𝑊–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ 𝐹:𝑊–1-1-onto→{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑠‘0) = 𝑋}) |
| 59 | 55, 58 | sylibr 234 |
1
⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝐹:𝑊–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁)) |