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Theorem uspgr2wlkeq 29434

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 1095	. . 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 29422	. . . 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 1116	. . 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 1128	. 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 13747	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
7	6	adantl 481	. . . . . . . . . . 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 2743	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 3599	. . . . . . . . . 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 451	. . . . . . . . 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 3140	. . . . . . . 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 7437	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐴)‘(𝑥 + 1)))
16		fvoveq1 7437	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((2^nd ‘𝐵)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1)))
17	15, 16	eqeq12d 2743	. . . . . . . . 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 3229	. . . . . . . 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 13669	. . . . . . . . . 10 ⊢ (0..^𝑁) ⊆ (0...𝑁)
21		ssralv 4046	. . . . . . . . . 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 3106	. . . . . . . . . . 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 3164	. . . . . . . . . . 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 415	. . . . . . . . 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 406	. . . . . . 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 412	. . . . 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 28965	. . . . . . . 8 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
34		eqid 2727	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
35		eqid 2727	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
36		eqid 2727	. . . . . . . . . 10 ⊢ (1^st ‘𝐴) = (1^st ‘𝐴)
37		eqid 2727	. . . . . . . . . 10 ⊢ (2^nd ‘𝐴) = (2^nd ‘𝐴)
38	34, 35, 36, 37	upgrwlkcompim 29431	. . . . . . . . 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 412	. . . . . . . 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 2727	. . . . . . . . . 10 ⊢ (1^st ‘𝐵) = (1^st ‘𝐵)
42		eqid 2727	. . . . . . . . . 10 ⊢ (2^nd ‘𝐵) = (2^nd ‘𝐵)
43	34, 35, 41, 42	upgrwlkcompim 29431	. . . . . . . . 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 412	. . . . . . . 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 7422	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1^st ‘𝐵)) = 𝑁 → (0..^(♯‘(1^st ‘𝐵))) = (0..^𝑁))
47	46	eqcoms 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (0..^(♯‘(1^st ‘𝐵))) = (0..^𝑁))
48	47	raleqdv 3320	. . . . . . . . . . . . . . . . 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 7422	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1^st ‘𝐴)) = 𝑁 → (0..^(♯‘(1^st ‘𝐴))) = (0..^𝑁))
50	49	eqcoms 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (0..^(♯‘(1^st ‘𝐴))) = (0..^𝑁))
51	50	raleqdv 3320	. . . . . . . . . . . . . . . . 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 3106	. . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . 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 3080	. . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 1133	. . . . . . . . . . . 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 1133	. . . . . . . . . 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 406	. . . . . . . . 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 415	. . . . . . . 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 607	. . . . . 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 1345	. . . . 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 2734	. . . . . . 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 28960	. . . . . . . . . . . 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 2728	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺) = (iEdg‘𝐺))
79		eqidd 2728	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → dom (iEdg‘𝐺) = dom (iEdg‘𝐺))
80		edgval 28836	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
81	80	eqcomi 2736	. . . . . . . . . . . . 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 257	. . . . . . . . . 10 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
85	84	3ad2ant1 1131	. . . . . . . . 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 480	. . . . . . . 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 29421	. . . . . . . . . . . . . . 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 29421	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (Walks‘𝐺) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)))
89		oveq2 7422	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (0..^𝑁) = (0..^(♯‘(1^st ‘𝐴))))
90	89	eleq2d 2814	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))))
91		wrdsymbcl 14495	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))
92	91	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . . 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 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 ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
98		oveq2 7422	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (0..^𝑁) = (0..^(♯‘(1^st ‘𝐵))))
99	98	eleq2d 2814	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))))
100		wrdsymbcl 14495	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))
101	100	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . . 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 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 ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
107	97, 106	jcad 512	. . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . 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 415	. . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 1128	. . . . . . . . . 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 406	. . . . . . . . 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 406	. . . . . . . 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 7261	. . . . . . . 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 684	. . . . . . 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 3162	. . . . 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 453	. . 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 561	. 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 311	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 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ⊆ wss 3944 {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 7414 1^st c1st 7983 2^nd c2nd 7984 0cc0 11124 1c1 11125 + caddc 11127 ...cfz 13502 ..^cfzo 13645 ♯chash 14307 Word cword 14482 Vtxcvtx 28783 iEdgciedg 28784 Edgcedg 28834 UPGraphcupgr 28867 USPGraphcuspgr 28935 Walkscwlks 29384

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-er 8716 df-map 8836 df-pm 8837 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-dju 9910 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-n0 12489 df-xnn0 12561 df-z 12575 df-uz 12839 df-fz 13503 df-fzo 13646 df-hash 14308 df-word 14483 df-edg 28835 df-uhgr 28845 df-upgr 28869 df-uspgr 28937 df-wlks 29387

This theorem is referenced by: uspgr2wlkeq2 29435 clwlkclwwlkf1 29794

W3C validator