uspgr2wlkeq - Mathbox for Alexander van der Vekens

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Theorem uspgr2wlkeq 40854

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 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 1043	. . 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 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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		1wlkeq 40838	. . . 4 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 1257	. . 3 ⊢ (((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 1072	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 12431	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
7	6	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1^st ‘𝐴))) ∧ 𝑁 = (#‘(1^st ‘𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁))
8		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 + 1) → ((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐴)‘(𝑥 + 1)))
9		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 + 1) → ((2^nd ‘𝐵)‘𝑦) = ((2^nd ‘𝐵)‘(𝑥 + 1)))
10	8, 9	eqeq12d 2625	. . . . . . . . . . . 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 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 3285	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 455	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 2948	. . . . . . . 8 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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		oveq1 6556	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦 + 1) = (𝑥 + 1))
16	15	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐴)‘(𝑥 + 1)))
17	15	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((2^nd ‘𝐵)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1)))
18	16, 17	eqeq12d 2625	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1)) ↔ ((2^nd ‘𝐴)‘(𝑥 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1))))
19	18	cbvralv 3147	. . . . . . . 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)))
20	14, 19	sylibr 223	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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)))
21		fzossfz 12357	. . . . . . . . . 10 ⊢ (0..^𝑁) ⊆ (0...𝑁)
22		ssralv 3629	. . . . . . . . . 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)))
23	21, 22	mp1i 13	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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)))
24		r19.26 3046	. . . . . . . . . . 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))))
25		preq12 4214	. . . . . . . . . . . . 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))})
26	25	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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))}))
27	26	ralimdv 2946	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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))}))
28	24, 27	syl5bir 232	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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	28	expd 451	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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	23, 29	syld 46	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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	30	imp 444	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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))}))
32	20, 31	mpd 15	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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	32	ex 449	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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))}))
34		uspgrupgr 40406	. . . . . . . 8 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )
35		eqid 2610	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
36		eqid 2610	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
37		eqid 2610	. . . . . . . . . 10 ⊢ (1^st ‘𝐴) = (1^st ‘𝐴)
38		eqid 2610	. . . . . . . . . 10 ⊢ (2^nd ‘𝐴) = (2^nd ‘𝐴)
39	35, 36, 37, 38	upgr1wlkcompim 40851	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ (1Walks‘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	39	ex 449	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (𝐴 ∈ (1Walks‘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	34, 40	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → (𝐴 ∈ (1Walks‘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))})))
42		eqid 2610	. . . . . . . . . 10 ⊢ (1^st ‘𝐵) = (1^st ‘𝐵)
43		eqid 2610	. . . . . . . . . 10 ⊢ (2^nd ‘𝐵) = (2^nd ‘𝐵)
44	35, 36, 42, 43	upgr1wlkcompim 40851	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐵 ∈ (1Walks‘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	44	ex 449	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (𝐵 ∈ (1Walks‘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	34, 45	syl 17	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → (𝐵 ∈ (1Walks‘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))})))
47		oveq2 6557	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘(1^st ‘𝐵)) = 𝑁 → (0..^(#‘(1^st ‘𝐵))) = (0..^𝑁))
48	47	eqcoms 2618	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (#‘(1^st ‘𝐵)) → (0..^(#‘(1^st ‘𝐵))) = (0..^𝑁))
49	48	raleqdv 3121	. . . . . . . . . . . . . . . . 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))}))
50		oveq2 6557	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘(1^st ‘𝐴)) = 𝑁 → (0..^(#‘(1^st ‘𝐴))) = (0..^𝑁))
51	50	eqcoms 2618	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (#‘(1^st ‘𝐴)) → (0..^(#‘(1^st ‘𝐴))) = (0..^𝑁))
52	51	raleqdv 3121	. . . . . . . . . . . . . . . . 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))}))
53	49, 52	bi2anan9r 914	. . . . . . . . . . . . . . . 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))})))
54		r19.26 3046	. . . . . . . . . . . . . . . . 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))}))
55		eqeq2 2621	. . . . . . . . . . . . . . . . . . . . 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))}))
56		eqeq2 2621	. . . . . . . . . . . . . . . . . . . . . . 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))))
57	56	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . 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))))
58	57	biimpd 218	. . . . . . . . . . . . . . . . . . . . 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))))
59	55, 58	syl6bi 242	. . . . . . . . . . . . . . . . . . . 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)))))
60	59	com13 86	. . . . . . . . . . . . . . . . . . 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)))))
61	60	imp 444	. . . . . . . . . . . . . . . . . 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))))
62	61	ral2imi 2931	. . . . . . . . . . . . . . . . 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))))
63	54, 62	sylbir 224	. . . . . . . . . . . . . . . 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))))
64	53, 63	syl6bi 242	. . . . . . . . . . . . . . 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)))))
65	64	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)))))
66	65	ex 449	. . . . . . . . . . . . 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))))))
67	66	3ad2ant3 1077	. . . . . . . . . . . 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))))))
68	67	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))))))
69	68	3ad2ant3 1077	. . . . . . . . . 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))))))
70	69	imp 444	. . . . . . . . 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)))))
71	70	expd 451	. . . . . . . 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))))))
72	71	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)))))))
73	41, 46, 72	syl2and 499	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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	73	3imp1 1272	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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))))
75		eqcom 2617	. . . . . . 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)))
76	36	uspgrf1oedg 40403	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
77		f1of1 6049	. . . . . . . . . . . 12 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺))
78	76, 77	syl 17	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺))
79		eqidd 2611	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺) = (iEdg‘𝐺))
80		eqidd 2611	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → dom (iEdg‘𝐺) = dom (iEdg‘𝐺))
81		edgaval 25794	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
82	81	eqcomd 2616	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → ran (iEdg‘𝐺) = (Edg‘𝐺))
83	79, 80, 82	f1eq123d 6044	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USPGraph → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)))
84	78, 83	mpbird 246	. . . . . . . . . 10 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
85	84	3ad2ant1 1075	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1^st ‘𝐴))) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
86	85	adantr 480	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1^st ‘𝐴))) ∧ 𝑁 = (#‘(1^st ‘𝐵))) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
87	35, 36, 37, 38	1wlkelwrd 40837	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (1Walks‘𝐺) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(#‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)))
88	35, 36, 42, 43	1wlkelwrd 40837	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (1Walks‘𝐺) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(#‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)))
89		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 = (#‘(1^st ‘𝐴)) → (0..^𝑁) = (0..^(#‘(1^st ‘𝐴))))
90	89	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 = (#‘(1^st ‘𝐴)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘(1^st ‘𝐴)))))
91		wrdsymbcl 13173	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(#‘(1^st ‘𝐴)))) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))
92	91	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(#‘(1^st ‘𝐴))) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
93	90, 92	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . 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 444	. . . . . . . . . . . . . . . . . . . . . . 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 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 = (#‘(1^st ‘𝐵)) → (0..^𝑁) = (0..^(#‘(1^st ‘𝐵))))
99	98	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 = (#‘(1^st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘(1^st ‘𝐵)))))
100		wrdsymbcl 13173	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(#‘(1^st ‘𝐵)))) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))
101	100	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(#‘(1^st ‘𝐵))) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
102	99, 101	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . 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 444	. . . . . . . . . . . . . . . . . . . . . . 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 554	. . . . . . . . . . . . . . . . . . . 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 449	. . . . . . . . . . . . . . . . . . 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 444	. . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 451	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 451	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 444	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 1072	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 444	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 444	. . . . . . . 8 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 6418	. . . . . . . 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 872	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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	75, 121	syl5bi 231	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 2945	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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	33, 74, 123	3syld 58	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 627	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 664	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 299	1 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 {cpr 4127 dom cdm 5038 ran crn 5039 ⟶wf 5800 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 0cc0 9815 1c1 9816 + caddc 9818 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 Vtxcvtx 25673 iEdgciedg 25674 UPGraph cupgr 25747 Edgcedga 25792 USPGraph cuspgr 40378 1Walksc1wlks 40796

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-uhgr 25724 df-upgr 25749 df-edga 25793 df-uspgr 40380 df-1wlks 40800 df-wlks 40801

This theorem is referenced by: uspgr2wlkeq2 40855 clwlksf1clwwlk 41276

W3C validator