uspgr2wlkeq - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > uspgr2wlkeq		Structured version Visualization version GIF version

Theorem uspgr2wlkeq 28892

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 1097	. . 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 28880	. . . 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 1118	. . 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 1130	. 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 13725	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
7	6	adantl 482	. . . . . . . . . . 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 6888	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 + 1) → ((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐴)‘(𝑥 + 1)))
9		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 + 1) → ((2^nd ‘𝐵)‘𝑦) = ((2^nd ‘𝐵)‘(𝑥 + 1)))
10	8, 9	eqeq12d 2748	. . . . . . . . . . . 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 482	. . . . . . . . . . 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 3604	. . . . . . . . . 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 452	. . . . . . . . 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 3145	. . . . . . . 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 7428	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐴)‘(𝑥 + 1)))
16		fvoveq1 7428	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((2^nd ‘𝐵)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1)))
17	15, 16	eqeq12d 2748	. . . . . . . . 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 3234	. . . . . . . 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 13647	. . . . . . . . . 10 ⊢ (0..^𝑁) ⊆ (0...𝑁)
21		ssralv 4049	. . . . . . . . . 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 3111	. . . . . . . . . . 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 4738	. . . . . . . . . . . . 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 3169	. . . . . . . . . . 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 416	. . . . . . . . 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 407	. . . . . . 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 413	. . . . 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 28425	. . . . . . . 8 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
34		eqid 2732	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
35		eqid 2732	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
36		eqid 2732	. . . . . . . . . 10 ⊢ (1^st ‘𝐴) = (1^st ‘𝐴)
37		eqid 2732	. . . . . . . . . 10 ⊢ (2^nd ‘𝐴) = (2^nd ‘𝐴)
38	34, 35, 36, 37	upgrwlkcompim 28889	. . . . . . . . 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 413	. . . . . . . 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 2732	. . . . . . . . . 10 ⊢ (1^st ‘𝐵) = (1^st ‘𝐵)
42		eqid 2732	. . . . . . . . . 10 ⊢ (2^nd ‘𝐵) = (2^nd ‘𝐵)
43	34, 35, 41, 42	upgrwlkcompim 28889	. . . . . . . . 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 413	. . . . . . . 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 7413	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1^st ‘𝐵)) = 𝑁 → (0..^(♯‘(1^st ‘𝐵))) = (0..^𝑁))
47	46	eqcoms 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (0..^(♯‘(1^st ‘𝐵))) = (0..^𝑁))
48	47	raleqdv 3325	. . . . . . . . . . . . . . . . 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 7413	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1^st ‘𝐴)) = 𝑁 → (0..^(♯‘(1^st ‘𝐴))) = (0..^𝑁))
50	49	eqcoms 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (0..^(♯‘(1^st ‘𝐴))) = (0..^𝑁))
51	50	raleqdv 3325	. . . . . . . . . . . . . . . . 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 638	. . . . . . . . . . . . . . . 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 3111	. . . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . . . . . 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	syl6bi 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 407	. . . . . . . . . . . . . . . . . 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 3085	. . . . . . . . . . . . . . . . 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	syl6bi 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 413	. . . . . . . . . . . . 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 1135	. . . . . . . . . . . 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 1135	. . . . . . . . . 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 407	. . . . . . . . 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 416	. . . . . . . 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 608	. . . . . 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 1347	. . . . 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 2739	. . . . . . 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 28422	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
76		f1of1 6829	. . . . . . . . . . . 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 2733	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺) = (iEdg‘𝐺))
79		eqidd 2733	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → dom (iEdg‘𝐺) = dom (iEdg‘𝐺))
80		edgval 28298	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
81	80	eqcomi 2741	. . . . . . . . . . . . 13 ⊢ ran (iEdg‘𝐺) = (Edg‘𝐺)
82	81	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → ran (iEdg‘𝐺) = (Edg‘𝐺))
83	78, 79, 82	f1eq123d 6822	. . . . . . . . . . 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 1133	. . . . . . . . 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 481	. . . . . . . 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 28879	. . . . . . . . . . . . . . 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 28879	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (Walks‘𝐺) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)))
89		oveq2 7413	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (0..^𝑁) = (0..^(♯‘(1^st ‘𝐴))))
90	89	eleq2d 2819	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))))
91		wrdsymbcl 14473	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))
92	91	expcom 414	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(♯‘(1^st ‘𝐴))) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
93	90, 92	syl6bi 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 481	. . . . . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . 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 7413	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (0..^𝑁) = (0..^(♯‘(1^st ‘𝐵))))
99	98	eleq2d 2819	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))))
100		wrdsymbcl 14473	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))
101	100	expcom 414	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(♯‘(1^st ‘𝐵))) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
102	99, 101	syl6bi 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 482	. . . . . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . 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 513	. . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 416	. . . . . . . . . . . . 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 416	. . . . . . . . . . . 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 407	. . . . . . . . . . 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 1130	. . . . . . . . . 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 407	. . . . . . . . 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 407	. . . . . . . 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 7252	. . . . . . . 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 3167	. . . . 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 454	. . 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 562	. 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ⊆ wss 3947 {cpr 4629 dom cdm 5675 ran crn 5676 ⟶wf 6536 –1-1→wf1 6537 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7405 1^st c1st 7969 2^nd c2nd 7970 0cc0 11106 1c1 11107 + caddc 11109 ...cfz 13480 ..^cfzo 13623 ♯chash 14286 Word cword 14460 Vtxcvtx 28245 iEdgciedg 28246 Edgcedg 28296 UPGraphcupgr 28329 USPGraphcuspgr 28397 Walkscwlks 28842

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-edg 28297 df-uhgr 28307 df-upgr 28331 df-uspgr 28399 df-wlks 28845

This theorem is referenced by: uspgr2wlkeq2 28893 clwlkclwwlkf1 29252

W3C validator