Theorem uspgr2wlkeq 29682

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 = (♯‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘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 = (♯‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) ↔ ((𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦)))
2	1	a1i 11	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → ((𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) ↔ ((𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦))))
3		wlkeq 29670	. . . 4 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦))))
4	3	3expa 1118	. . 3 ⊢ (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦))))
5	4	3adant1 1130	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦))))
6		fzofzp1 13814	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
7	6	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁))
8		fveq2 6920	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 + 1) → ((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐴)‘(𝑥 + 1)))
9		fveq2 6920	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 + 1) → ((2nd ‘𝐵)‘𝑦) = ((2nd ‘𝐵)‘(𝑥 + 1)))
10	8, 9	eqeq12d 2756	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 + 1) → (((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ↔ ((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1))))
11	10	adantl 481	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) ∧ 𝑦 = (𝑥 + 1)) → (((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ↔ ((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1))))
12	7, 11	rspcdv 3627	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) → ((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1))))
13	12	impancom 451	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) → (𝑥 ∈ (0..^𝑁) → ((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1))))
14	13	ralrimiv 3151	. . . . . . . 8 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) → ∀𝑥 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1)))
15		fvoveq1 7471	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐴)‘(𝑥 + 1)))
16		fvoveq1 7471	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((2nd ‘𝐵)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1)))
17	15, 16	eqeq12d 2756	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1)) ↔ ((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1))))
18	17	cbvralvw 3243	. . . . . . . 8 ⊢ (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1)) ↔ ∀𝑥 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑥 + 1)) = ((2nd ‘𝐵)‘(𝑥 + 1)))
19	14, 18	sylibr 234	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1)))
20		fzossfz 13735	. . . . . . . . . 10 ⊢ (0..^𝑁) ⊆ (0...𝑁)
21		ssralv 4077	. . . . . . . . . 10 ⊢ ((0..^𝑁) ⊆ (0...𝑁) → (∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)))
22	20, 21	mp1i 13	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)))
23		r19.26 3117	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ (0..^𝑁)(((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ∧ ((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1))) ↔ (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1))))
24		preq12 4760	. . . . . . . . . . . . 13 ⊢ ((((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ∧ ((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1))) → {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})
25	24	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → ((((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ∧ ((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1))) → {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
26	25	ralimdv 3175	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁)(((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ∧ ((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1))) → ∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
27	23, 26	biimtrrid 243	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → ((∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1))) → ∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
28	27	expd 415	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) → (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})))
29	22, 28	syld 47	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) → (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})))
30	29	imp 406	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) → (∀𝑦 ∈ (0..^𝑁)((2nd ‘𝐴)‘(𝑦 + 1)) = ((2nd ‘𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
31	19, 30	mpd 15	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})
32	31	ex 412	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
33		uspgrupgr 29213	. . . . . . . 8 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
34		eqid 2740	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
35		eqid 2740	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
36		eqid 2740	. . . . . . . . . 10 ⊢ (1st ‘𝐴) = (1st ‘𝐴)
37		eqid 2740	. . . . . . . . . 10 ⊢ (2nd ‘𝐴) = (2nd ‘𝐴)
38	34, 35, 36, 37	upgrwlkcompim 29679	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ (Walks‘𝐺)) → ((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐴):(0...(♯‘(1st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}))
39	38	ex 412	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (𝐴 ∈ (Walks‘𝐺) → ((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐴):(0...(♯‘(1st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))})))
40	33, 39	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → (𝐴 ∈ (Walks‘𝐺) → ((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐴):(0...(♯‘(1st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))})))
41		eqid 2740	. . . . . . . . . 10 ⊢ (1st ‘𝐵) = (1st ‘𝐵)
42		eqid 2740	. . . . . . . . . 10 ⊢ (2nd ‘𝐵) = (2nd ‘𝐵)
43	34, 35, 41, 42	upgrwlkcompim 29679	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐵 ∈ (Walks‘𝐺)) → ((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐵):(0...(♯‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
44	43	ex 412	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (𝐵 ∈ (Walks‘𝐺) → ((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐵):(0...(♯‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})))
45	33, 44	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → (𝐵 ∈ (Walks‘𝐺) → ((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐵):(0...(♯‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})))
46		oveq2 7456	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1st ‘𝐵)) = 𝑁 → (0..^(♯‘(1st ‘𝐵))) = (0..^𝑁))
47	46	eqcoms 2748	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘(1st ‘𝐵)) → (0..^(♯‘(1st ‘𝐵))) = (0..^𝑁))
48	47	raleqdv 3334	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘(1st ‘𝐵)) → (∀𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ↔ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
49		oveq2 7456	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1st ‘𝐴)) = 𝑁 → (0..^(♯‘(1st ‘𝐴))) = (0..^𝑁))
50	49	eqcoms 2748	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘(1st ‘𝐴)) → (0..^(♯‘(1st ‘𝐴))) = (0..^𝑁))
51	50	raleqdv 3334	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘(1st ‘𝐴)) → (∀𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} ↔ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}))
52	48, 51	bi2anan9r 638	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → ((∀𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) ↔ (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))})))
53		r19.26 3117	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ (0..^𝑁)(((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) ↔ (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}))
54		eqeq2 2752	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} ↔ ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}))
55		eqeq2 2752	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) → (((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ↔ ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))
56	55	eqcoms 2748	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ↔ ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))
57	56	biimpd 229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))
58	54, 57	biimtrdi 253	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))))
59	58	com13 88	. . . . . . . . . . . . . . . . . . 19 ⊢ (((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} → ({((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))))
60	59	imp 406	. . . . . . . . . . . . . . . . . 18 ⊢ ((((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) → ({((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))
61	60	ral2imi 3091	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ (0..^𝑁)(((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))
62	53, 61	sylbir 235	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))
63	52, 62	biimtrdi 253	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → ((∀𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))))
64	63	com12 32	. . . . . . . . . . . . . 14 ⊢ ((∀𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) → ((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))))
65	64	ex 412	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → (∀𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} → ((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))))
66	65	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ (((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐵):(0...(♯‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} → ((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))))
67	66	com12 32	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} → (((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐵):(0...(♯‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}) → ((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))))
68	67	3ad2ant3 1135	. . . . . . . . . 10 ⊢ (((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐴):(0...(♯‘(1st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) → (((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐵):(0...(♯‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))}) → ((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))))
69	68	imp 406	. . . . . . . . 9 ⊢ ((((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐴):(0...(♯‘(1st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) ∧ ((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐵):(0...(♯‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})) → ((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))))
70	69	expd 415	. . . . . . . 8 ⊢ ((((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐴):(0...(♯‘(1st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) ∧ ((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐵):(0...(♯‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})) → (𝑁 = (♯‘(1st ‘𝐴)) → (𝑁 = (♯‘(1st ‘𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))))
71	70	a1i 11	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → ((((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐴):(0...(♯‘(1st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = {((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))}) ∧ ((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐵):(0...(♯‘(1st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))})) → (𝑁 = (♯‘(1st ‘𝐴)) → (𝑁 = (♯‘(1st ‘𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))))))
72	40, 45, 71	syl2and 607	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (𝑁 = (♯‘(1st ‘𝐴)) → (𝑁 = (♯‘(1st ‘𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)))))))
73	72	3imp1 1347	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2nd ‘𝐴)‘𝑦), ((2nd ‘𝐴)‘(𝑦 + 1))} = {((2nd ‘𝐵)‘𝑦), ((2nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦))))
74		eqcom 2747	. . . . . . 7 ⊢ (((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) ↔ ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)))
75	35	uspgrf1oedg 29208	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
76		f1of1 6861	. . . . . . . . . . . 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 2741	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺) = (iEdg‘𝐺))
79		eqidd 2741	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → dom (iEdg‘𝐺) = dom (iEdg‘𝐺))
80		edgval 29084	. . . . . . . . . . . . . 14 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
81	80	eqcomi 2749	. . . . . . . . . . . . 13 ⊢ ran (iEdg‘𝐺) = (Edg‘𝐺)
82	81	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → ran (iEdg‘𝐺) = (Edg‘𝐺))
83	78, 79, 82	f1eq123d 6854	. . . . . . . . . . 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 1133	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘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 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
87	34, 35, 36, 37	wlkelwrd 29669	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (Walks‘𝐺) → ((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐴):(0...(♯‘(1st ‘𝐴)))⟶(Vtx‘𝐺)))
88	34, 35, 41, 42	wlkelwrd 29669	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (Walks‘𝐺) → ((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐵):(0...(♯‘(1st ‘𝐵)))⟶(Vtx‘𝐺)))
89		oveq2 7456	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 = (♯‘(1st ‘𝐴)) → (0..^𝑁) = (0..^(♯‘(1st ‘𝐴))))
90	89	eleq2d 2830	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 = (♯‘(1st ‘𝐴)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))))
91		wrdsymbcl 14575	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1st ‘𝐴)))) → ((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))
92	91	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(♯‘(1st ‘𝐴))) → ((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → ((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
93	90, 92	biimtrdi 253	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 = (♯‘(1st ‘𝐴)) → (𝑦 ∈ (0..^𝑁) → ((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → ((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))))
94	93	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → ((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))))
95	94	imp 406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → ((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
96	95	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
97	96	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (1st ‘𝐴) ∈ Word dom (iEdg‘𝐺)) → (((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
98		oveq2 7456	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 = (♯‘(1st ‘𝐵)) → (0..^𝑁) = (0..^(♯‘(1st ‘𝐵))))
99	98	eleq2d 2830	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 = (♯‘(1st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))))
100		wrdsymbcl 14575	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1st ‘𝐵)))) → ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))
101	100	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(♯‘(1st ‘𝐵))) → ((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
102	99, 101	biimtrdi 253	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 = (♯‘(1st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → ((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
103	102	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
104	103	imp 406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
105	104	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
106	105	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (1st ‘𝐴) ∈ Word dom (iEdg‘𝐺)) → (((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
107	97, 106	jcad 512	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (1st ‘𝐴) ∈ Word dom (iEdg‘𝐺)) → (((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
108	107	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
109	108	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐵):(0...(♯‘(1st ‘𝐵)))⟶(Vtx‘𝐺)) → ((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
110	109	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐵):(0...(♯‘(1st ‘𝐵)))⟶(Vtx‘𝐺)) → (((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
111	110	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐴):(0...(♯‘(1st ‘𝐴)))⟶(Vtx‘𝐺)) → (((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐵):(0...(♯‘(1st ‘𝐵)))⟶(Vtx‘𝐺)) → (((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
112	111	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((((1st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐴):(0...(♯‘(1st ‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝐵):(0...(♯‘(1st ‘𝐵)))⟶(Vtx‘𝐺))) → (((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
113	87, 88, 112	syl2an 595	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
114	113	expd 415	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → ((𝑁 = (♯‘(1st ‘𝐴)) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
115	114	expd 415	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (𝑁 = (♯‘(1st ‘𝐴)) → (𝑁 = (♯‘(1st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))))
116	115	imp 406	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → (𝑁 = (♯‘(1st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
117	116	3adant1 1130	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → (𝑁 = (♯‘(1st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
118	117	imp 406	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
119	118	imp 406	. . . . . . . 8 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
120		f1veqaeq 7294	. . . . . . . 8 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺) ∧ (((1st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))) → (((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) → ((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦)))
121	86, 119, 120	syl2an2r 684	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) → ((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦)))
122	74, 121	biimtrid 242	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) → ((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦)))
123	122	ralimdva 3173	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1st ‘𝐴)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦)))
124	32, 73, 123	3syld 60	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) ∧ 𝑁 = (♯‘(1st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦)))
125	124	expimpd 453	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → ((𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦)))
126	125	pm4.71d 561	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → ((𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) ↔ ((𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑦) = ((1st ‘𝐵)‘𝑦))))
127	2, 5, 126	3bitr4d 311	1 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑦) = ((2nd ‘𝐵)‘𝑦))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976  {cpr 4650  dom cdm 5700  ran crn 5701  ⟶wf 6569  –1-1→wf1 6570  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  0cc0 11184  1c1 11185   + caddc 11187  ...cfz 13567  ..^cfzo 13711  ♯chash 14379  Word cword 14562  Vtxcvtx 29031  iEdgciedg 29032  Edgcedg 29082  UPGraphcupgr 29115  USPGraphcuspgr 29183  Walkscwlks 29632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-edg 29083  df-uhgr 29093  df-upgr 29117  df-uspgr 29185  df-wlks 29635

This theorem is referenced by:  uspgr2wlkeq2  29683  clwlkclwwlkf1  30042