uspgr2wlkeq - Metamath Proof Explorer

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Theorem uspgr2wlkeq 29502

Description: Conditions for two walks within the same simple pseudograph being the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 3-Jul-2018.) (Revised by AV, 14-Apr-2021.)

**Assertion**
Ref	Expression
uspgr2wlkeq	⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦))))

Distinct variable groups: 𝑦,𝐴 𝑦,𝐵 𝑦,𝐺 𝑦,𝑁

Proof of Theorem uspgr2wlkeq Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3anan32 1094	. . 3 ⊢ ((𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) ↔ ((𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦)))
2	1	a1i 11	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → ((𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) ↔ ((𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦))))
3		wlkeq 29490	. . . 4 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦))))
4	3	3expa 1115	. . 3 ⊢ (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦))))
5	4	3adant1 1127	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦))))
6		fzofzp1 13759	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
7	6	adantl 480	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁))
8		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 + 1) → ((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐴)‘(𝑥 + 1)))
9		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 + 1) → ((2^nd ‘𝐵)‘𝑦) = ((2^nd ‘𝐵)‘(𝑥 + 1)))
10	8, 9	eqeq12d 2741	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 + 1) → (((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ↔ ((2^nd ‘𝐴)‘(𝑥 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1))))
11	10	adantl 480	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) ∧ 𝑦 = (𝑥 + 1)) → (((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ↔ ((2^nd ‘𝐴)‘(𝑥 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1))))
12	7, 11	rspcdv 3594	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) → ((2^nd ‘𝐴)‘(𝑥 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1))))
13	12	impancom 450	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) → (𝑥 ∈ (0..^𝑁) → ((2^nd ‘𝐴)‘(𝑥 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1))))
14	13	ralrimiv 3135	. . . . . . . 8 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) → ∀𝑥 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑥 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1)))
15		fvoveq1 7438	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐴)‘(𝑥 + 1)))
16		fvoveq1 7438	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((2^nd ‘𝐵)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1)))
17	15, 16	eqeq12d 2741	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1)) ↔ ((2^nd ‘𝐴)‘(𝑥 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1))))
18	17	cbvralvw 3225	. . . . . . . 8 ⊢ (∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1)) ↔ ∀𝑥 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑥 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1)))
19	14, 18	sylibr 233	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1)))
20		fzossfz 13681	. . . . . . . . . 10 ⊢ (0..^𝑁) ⊆ (0...𝑁)
21		ssralv 4041	. . . . . . . . . 10 ⊢ ((0..^𝑁) ⊆ (0...𝑁) → (∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)))
22	20, 21	mp1i 13	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)))
23		r19.26 3101	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ (0..^𝑁)(((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ∧ ((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1))) ↔ (∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1))))
24		preq12 4735	. . . . . . . . . . . . 13 ⊢ ((((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ∧ ((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1))) → {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})
25	24	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → ((((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ∧ ((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1))) → {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
26	25	ralimdv 3159	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁)(((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ∧ ((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1))) → ∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
27	23, 26	biimtrrid 242	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → ((∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1))) → ∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
28	27	expd 414	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) → (∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})))
29	22, 28	syld 47	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) → (∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})))
30	29	imp 405	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) → (∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
31	19, 30	mpd 15	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})
32	31	ex 411	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
33		uspgrupgr 29033	. . . . . . . 8 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
34		eqid 2725	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
35		eqid 2725	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
36		eqid 2725	. . . . . . . . . 10 ⊢ (1^st ‘𝐴) = (1^st ‘𝐴)
37		eqid 2725	. . . . . . . . . 10 ⊢ (2^nd ‘𝐴) = (2^nd ‘𝐴)
38	34, 35, 36, 37	upgrwlkcompim 29499	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ (Walks‘𝐺)) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}))
39	38	ex 411	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (𝐴 ∈ (Walks‘𝐺) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))})))
40	33, 39	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → (𝐴 ∈ (Walks‘𝐺) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))})))
41		eqid 2725	. . . . . . . . . 10 ⊢ (1^st ‘𝐵) = (1^st ‘𝐵)
42		eqid 2725	. . . . . . . . . 10 ⊢ (2^nd ‘𝐵) = (2^nd ‘𝐵)
43	34, 35, 41, 42	upgrwlkcompim 29499	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐵 ∈ (Walks‘𝐺)) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
44	43	ex 411	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (𝐵 ∈ (Walks‘𝐺) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})))
45	33, 44	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → (𝐵 ∈ (Walks‘𝐺) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})))
46		oveq2 7423	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1^st ‘𝐵)) = 𝑁 → (0..^(♯‘(1^st ‘𝐵))) = (0..^𝑁))
47	46	eqcoms 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (0..^(♯‘(1^st ‘𝐵))) = (0..^𝑁))
48	47	raleqdv 3315	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ↔ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
49		oveq2 7423	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1^st ‘𝐴)) = 𝑁 → (0..^(♯‘(1^st ‘𝐴))) = (0..^𝑁))
50	49	eqcoms 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (0..^(♯‘(1^st ‘𝐴))) = (0..^𝑁))
51	50	raleqdv 3315	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} ↔ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}))
52	48, 51	bi2anan9r 637	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → ((∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) ↔ (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))})))
53		r19.26 3101	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ (0..^𝑁)(((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) ↔ (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}))
54		eqeq2 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} ↔ ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
55		eqeq2 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) → (((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ↔ ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))
56	55	eqcoms 2733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ↔ ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))
57	56	biimpd 228	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))
58	54, 57	biimtrdi 252	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)))))
59	58	com13 88	. . . . . . . . . . . . . . . . . . 19 ⊢ (((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} → ({((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)))))
60	59	imp 405	. . . . . . . . . . . . . . . . . 18 ⊢ ((((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) → ({((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))
61	60	ral2imi 3075	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ (0..^𝑁)(((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))
62	53, 61	sylbir 234	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))
63	52, 62	biimtrdi 252	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → ((∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)))))
64	63	com12 32	. . . . . . . . . . . . . 14 ⊢ ((∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) → ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)))))
65	64	ex 411	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → (∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} → ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))))
66	65	3ad2ant3 1132	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} → ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))))
67	66	com12 32	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} → (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}) → ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))))
68	67	3ad2ant3 1132	. . . . . . . . . 10 ⊢ (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) → (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}) → ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))))
69	68	imp 405	. . . . . . . . 9 ⊢ ((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})) → ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)))))
70	69	expd 414	. . . . . . . 8 ⊢ ((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})) → (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑁 = (♯‘(1^st ‘𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))))
71	70	a1i 11	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → ((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})) → (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑁 = (♯‘(1^st ‘𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)))))))
72	40, 45, 71	syl2and 606	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑁 = (♯‘(1^st ‘𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)))))))
73	72	3imp1 1344	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))
74		eqcom 2732	. . . . . . 7 ⊢ (((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) ↔ ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)))
75	35	uspgrf1oedg 29028	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
76		f1of1 6832	. . . . . . . . . . . 12 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺))
77	75, 76	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺))
78		eqidd 2726	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺) = (iEdg‘𝐺))
79		eqidd 2726	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → dom (iEdg‘𝐺) = dom (iEdg‘𝐺))
80		edgval 28904	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
81	80	eqcomi 2734	. . . . . . . . . . . . 13 ⊢ ran (iEdg‘𝐺) = (Edg‘𝐺)
82	81	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → ran (iEdg‘𝐺) = (Edg‘𝐺))
83	78, 79, 82	f1eq123d 6825	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USPGraph → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)))
84	77, 83	mpbird 256	. . . . . . . . . 10 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
85	84	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
86	85	adantr 479	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
87	34, 35, 36, 37	wlkelwrd 29489	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (Walks‘𝐺) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)))
88	34, 35, 41, 42	wlkelwrd 29489	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (Walks‘𝐺) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)))
89		oveq2 7423	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (0..^𝑁) = (0..^(♯‘(1^st ‘𝐴))))
90	89	eleq2d 2811	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))))
91		wrdsymbcl 14507	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))
92	91	expcom 412	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(♯‘(1^st ‘𝐴))) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
93	90, 92	biimtrdi 252	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑦 ∈ (0..^𝑁) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))))
94	93	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))))
95	94	imp 405	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
96	95	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
97	96	adantl 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺)) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
98		oveq2 7423	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (0..^𝑁) = (0..^(♯‘(1^st ‘𝐵))))
99	98	eleq2d 2811	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))))
100		wrdsymbcl 14507	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))
101	100	expcom 412	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(♯‘(1^st ‘𝐵))) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
102	99, 101	biimtrdi 252	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
103	102	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
104	103	imp 405	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
105	104	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
106	105	adantr 479	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺)) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
107	97, 106	jcad 511	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺)) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
108	107	ex 411	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
109	108	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
110	109	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
111	110	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) → (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
112	111	imp 405	. . . . . . . . . . . . . . 15 ⊢ ((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺))) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
113	87, 88, 112	syl2an 594	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
114	113	expd 414	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
115	114	expd 414	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑁 = (♯‘(1^st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))))
116	115	imp 405	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (𝑁 = (♯‘(1^st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
117	116	3adant1 1127	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (𝑁 = (♯‘(1^st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
118	117	imp 405	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
119	118	imp 405	. . . . . . . 8 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
120		f1veqaeq 7262	. . . . . . . 8 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺) ∧ (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))) → (((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) → ((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦)))
121	86, 119, 120	syl2an2r 683	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) → ((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦)))
122	74, 121	biimtrid 241	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) → ((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦)))
123	122	ralimdva 3157	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦)))
124	32, 73, 123	3syld 60	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦)))
125	124	expimpd 452	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → ((𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦)))
126	125	pm4.71d 560	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → ((𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) ↔ ((𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦))))
127	2, 5, 126	3bitr4d 310	1 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ⊆ wss 3940 {cpr 4626 dom cdm 5672 ran crn 5673 ⟶wf 6538 –1-1→wf1 6539 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7415 1^st c1st 7987 2^nd c2nd 7988 0cc0 11136 1c1 11137 + caddc 11139 ...cfz 13514 ..^cfzo 13657 ♯chash 14319 Word cword 14494 Vtxcvtx 28851 iEdgciedg 28852 Edgcedg 28902 UPGraphcupgr 28935 USPGraphcuspgr 29003 Walkscwlks 29452

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-oadd 8487 df-er 8721 df-map 8843 df-pm 8844 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-dju 9922 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-n0 12501 df-xnn0 12573 df-z 12587 df-uz 12851 df-fz 13515 df-fzo 13658 df-hash 14320 df-word 14495 df-edg 28903 df-uhgr 28913 df-upgr 28937 df-uspgr 29005 df-wlks 29455

This theorem is referenced by: uspgr2wlkeq2 29503 clwlkclwwlkf1 29862

W3C validator