Theorem usgreg2spot 26594

Description: In a finite k-regular graph the set of all paths of length two is the union of all the paths of length 2 over the vertices which are in the middle of such a path. (Contributed by Alexander van der Vekens, 10-Mar-2018.)

Hypothesis

Ref	Expression
usgreghash2spot.m	⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑎)})

Assertion

Ref	Expression
usgreg2spot	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝑉 2SPathsOt 𝐸) = ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥)))

Distinct variable groups:   𝑡,𝐸,𝑥,𝑎   𝑉,𝑎,𝑡,𝑥   𝐸,𝑎,𝑣,𝑡   𝑣,𝑉,𝑎   𝑥,𝐾   𝑣,𝑀   𝑥,𝑣

Allowed substitution hints:   𝐾(𝑣,𝑡,𝑎)   𝑀(𝑥,𝑡,𝑎)

Proof of Theorem usgreg2spot

Dummy variables 𝑝 𝑦 𝑧 𝑓 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrav 25867	. . . . . . . . . 10 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2		el2pthsot 26408	. . . . . . . . . 10 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑦 ∈ 𝑉 ∃𝑥 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓∃𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2))))))
3	1, 2	syl 17	. . . . . . . . 9 ⊢ (𝑉 USGrph 𝐸 → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑦 ∈ 𝑉 ∃𝑥 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓∃𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2))))))
4		simpr 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉)
5	4	ad2antrr 758	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) ∧ 𝑧 ∈ 𝑉) → 𝑦 ∈ 𝑉)
6		simpr 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
7	6	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) ∧ 𝑧 ∈ 𝑉) → 𝑥 ∈ 𝑉)
8		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) ∧ 𝑧 ∈ 𝑉) → 𝑧 ∈ 𝑉)
9	5, 7, 8	3jca 1235	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) ∧ 𝑧 ∈ 𝑉) → (𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉))
10		ot2ndg 7074	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (2nd ‘(1st ‘⟨𝑦, 𝑥, 𝑧⟩)) = 𝑥)
11	9, 10	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) ∧ 𝑧 ∈ 𝑉) → (2nd ‘(1st ‘⟨𝑦, 𝑥, 𝑧⟩)) = 𝑥)
12		eqidd 2611	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) ∧ 𝑧 ∈ 𝑉) → ⟨𝑦, 𝑥, 𝑧⟩ = ⟨𝑦, 𝑥, 𝑧⟩)
13		otel3xp 5077	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨𝑦, 𝑥, 𝑧⟩ = ⟨𝑦, 𝑥, 𝑧⟩ ∧ (𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ⟨𝑦, 𝑥, 𝑧⟩ ∈ ((𝑉 × 𝑉) × 𝑉))
14	12, 9, 13	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) ∧ 𝑧 ∈ 𝑉) → ⟨𝑦, 𝑥, 𝑧⟩ ∈ ((𝑉 × 𝑉) × 𝑉))
15	11, 14	jca 553	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) ∧ 𝑧 ∈ 𝑉) → ((2nd ‘(1st ‘⟨𝑦, 𝑥, 𝑧⟩)) = 𝑥 ∧ ⟨𝑦, 𝑥, 𝑧⟩ ∈ ((𝑉 × 𝑉) × 𝑉)))
16		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ → (1st ‘𝑝) = (1st ‘⟨𝑦, 𝑥, 𝑧⟩))
17	16	fveq2d 6107	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ → (2nd ‘(1st ‘𝑝)) = (2nd ‘(1st ‘⟨𝑦, 𝑥, 𝑧⟩)))
18	17	eqeq1d 2612	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ → ((2nd ‘(1st ‘𝑝)) = 𝑥 ↔ (2nd ‘(1st ‘⟨𝑦, 𝑥, 𝑧⟩)) = 𝑥))
19		eleq1 2676	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ → (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ↔ ⟨𝑦, 𝑥, 𝑧⟩ ∈ ((𝑉 × 𝑉) × 𝑉)))
20	18, 19	anbi12d 743	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ → (((2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) ↔ ((2nd ‘(1st ‘⟨𝑦, 𝑥, 𝑧⟩)) = 𝑥 ∧ ⟨𝑦, 𝑥, 𝑧⟩ ∈ ((𝑉 × 𝑉) × 𝑉))))
21	15, 20	syl5ibr 235	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ → ((((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) ∧ 𝑧 ∈ 𝑉) → ((2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
22	21	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓∃𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2)))) → ((((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) ∧ 𝑧 ∈ 𝑉) → ((2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
23	22	com12 32	. . . . . . . . . . . . 13 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) ∧ 𝑧 ∈ 𝑉) → ((𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓∃𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2)))) → ((2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
24	23	rexlimdva 3013	. . . . . . . . . . . 12 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) ∧ 𝑥 ∈ 𝑉) → (∃𝑧 ∈ 𝑉 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓∃𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2)))) → ((2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
25	24	reximdva 3000	. . . . . . . . . . 11 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) → (∃𝑥 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓∃𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2)))) → ∃𝑥 ∈ 𝑉 ((2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
26		r19.41v 3070	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ 𝑉 ((2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) ↔ (∃𝑥 ∈ 𝑉 (2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)))
27	25, 26	syl6ib 240	. . . . . . . . . 10 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑦 ∈ 𝑉) → (∃𝑥 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓∃𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2)))) → (∃𝑥 ∈ 𝑉 (2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
28	27	rexlimdva 3013	. . . . . . . . 9 ⊢ (𝑉 USGrph 𝐸 → (∃𝑦 ∈ 𝑉 ∃𝑥 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑝 = ⟨𝑦, 𝑥, 𝑧⟩ ∧ ∃𝑓∃𝑞(𝑓(𝑉 SPaths 𝐸)𝑞 ∧ (#‘𝑓) = 2 ∧ (𝑦 = (𝑞‘0) ∧ 𝑥 = (𝑞‘1) ∧ 𝑧 = (𝑞‘2)))) → (∃𝑥 ∈ 𝑉 (2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
29	3, 28	sylbid 229	. . . . . . . 8 ⊢ (𝑉 USGrph 𝐸 → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) → (∃𝑥 ∈ 𝑉 (2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
30	29	ad2antrr 758	. . . . . . 7 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) → (∃𝑥 ∈ 𝑉 (2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉))))
31	30	pm4.71rd 665	. . . . . 6 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ↔ ((∃𝑥 ∈ 𝑉 (2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸))))
32		anass 679	. . . . . . . . 9 ⊢ (((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ∧ (2nd ‘(1st ‘𝑝)) = 𝑥) ↔ (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑝)) = 𝑥)))
33	32	bicomi 213	. . . . . . . 8 ⊢ ((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑝)) = 𝑥)) ↔ ((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ∧ (2nd ‘(1st ‘𝑝)) = 𝑥))
34	33	rexbii 3023	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑉 (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑝)) = 𝑥)) ↔ ∃𝑥 ∈ 𝑉 ((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ∧ (2nd ‘(1st ‘𝑝)) = 𝑥))
35		ancom 465	. . . . . . . 8 ⊢ (((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ∧ ∃𝑥 ∈ 𝑉 (2nd ‘(1st ‘𝑝)) = 𝑥) ↔ (∃𝑥 ∈ 𝑉 (2nd ‘(1st ‘𝑝)) = 𝑥 ∧ (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸))))
36		r19.42v 3073	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝑉 ((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ∧ (2nd ‘(1st ‘𝑝)) = 𝑥) ↔ ((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ∧ ∃𝑥 ∈ 𝑉 (2nd ‘(1st ‘𝑝)) = 𝑥))
37		anass 679	. . . . . . . 8 ⊢ (((∃𝑥 ∈ 𝑉 (2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ↔ (∃𝑥 ∈ 𝑉 (2nd ‘(1st ‘𝑝)) = 𝑥 ∧ (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸))))
38	35, 36, 37	3bitr4i 291	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝑉 ((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)) ∧ (2nd ‘(1st ‘𝑝)) = 𝑥) ↔ ((∃𝑥 ∈ 𝑉 (2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)))
39	34, 38	bitri 263	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑉 (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑝)) = 𝑥)) ↔ ((∃𝑥 ∈ 𝑉 (2nd ‘(1st ‘𝑝)) = 𝑥 ∧ 𝑝 ∈ ((𝑉 × 𝑉) × 𝑉)) ∧ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)))
40	31, 39	syl6bbr 277	. . . . 5 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑥 ∈ 𝑉 (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑝)) = 𝑥))))
41		simpr 476	. . . . . . . . 9 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
42		3xpexg 6859	. . . . . . . . . . 11 ⊢ (𝑉 ∈ Fin → ((𝑉 × 𝑉) × 𝑉) ∈ V)
43		rabexg 4739	. . . . . . . . . . 11 ⊢ (((𝑉 × 𝑉) × 𝑉) ∈ V → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)} ∈ V)
44	42, 43	syl 17	. . . . . . . . . 10 ⊢ (𝑉 ∈ Fin → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)} ∈ V)
45	44	ad3antlr 763	. . . . . . . . 9 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑥 ∈ 𝑉) → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)} ∈ V)
46		eqeq2 2621	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → ((2nd ‘(1st ‘𝑡)) = 𝑎 ↔ (2nd ‘(1st ‘𝑡)) = 𝑥))
47	46	anbi2d 736	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → ((𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑎) ↔ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)))
48	47	rabbidv 3164	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑎)} = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)})
49		usgreghash2spot.m	. . . . . . . . . 10 ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑎)})
50	48, 49	fvmptg 6189	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑉 ∧ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)} ∈ V) → (𝑀‘𝑥) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)})
51	41, 45, 50	syl2anc 691	. . . . . . . 8 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑥 ∈ 𝑉) → (𝑀‘𝑥) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)})
52	51	eleq2d 2673	. . . . . . 7 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑥 ∈ 𝑉) → (𝑝 ∈ (𝑀‘𝑥) ↔ 𝑝 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)}))
53		eleq1 2676	. . . . . . . . 9 ⊢ (𝑡 = 𝑝 → (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ↔ 𝑝 ∈ (𝑉 2SPathsOt 𝐸)))
54		fveq2 6103	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑝 → (1st ‘𝑡) = (1st ‘𝑝))
55	54	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑡 = 𝑝 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑝)))
56	55	eqeq1d 2612	. . . . . . . . 9 ⊢ (𝑡 = 𝑝 → ((2nd ‘(1st ‘𝑡)) = 𝑥 ↔ (2nd ‘(1st ‘𝑝)) = 𝑥))
57	53, 56	anbi12d 743	. . . . . . . 8 ⊢ (𝑡 = 𝑝 → ((𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥) ↔ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑝)) = 𝑥)))
58	57	elrab 3331	. . . . . . 7 ⊢ (𝑝 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑡)) = 𝑥)} ↔ (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑝)) = 𝑥)))
59	52, 58	syl6rbb 276	. . . . . 6 ⊢ ((((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) ∧ 𝑥 ∈ 𝑉) → ((𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑝)) = 𝑥)) ↔ 𝑝 ∈ (𝑀‘𝑥)))
60	59	rexbidva 3031	. . . . 5 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (∃𝑥 ∈ 𝑉 (𝑝 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st ‘𝑝)) = 𝑥)) ↔ ∃𝑥 ∈ 𝑉 𝑝 ∈ (𝑀‘𝑥)))
61	40, 60	bitrd 267	. . . 4 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑥 ∈ 𝑉 𝑝 ∈ (𝑀‘𝑥)))
62		eliun 4460	. . . 4 ⊢ (𝑝 ∈ ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥) ↔ ∃𝑥 ∈ 𝑉 𝑝 ∈ (𝑀‘𝑥))
63	61, 62	syl6bbr 277	. . 3 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑝 ∈ (𝑉 2SPathsOt 𝐸) ↔ 𝑝 ∈ ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥)))
64	63	eqrdv 2608	. 2 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) ∧ ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾) → (𝑉 2SPathsOt 𝐸) = ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥))
65	64	ex 449	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝑉 2SPathsOt 𝐸) = ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ⟨cotp 4133  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  Fincfn 7841  0cc0 9815  1c1 9816  2c2 10947  #chash 12979   USGrph cusg 25859   SPaths cspath 26029   2SPathsOt c2spthot 26383   VDeg cvdg 26420

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-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-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-ot 4134  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-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-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-usgra 25862  df-wlk 26036  df-trail 26037  df-pth 26038  df-spth 26039  df-wlkon 26042  df-spthon 26045  df-2spthonot 26387  df-2spthsot 26388

This theorem is referenced by:  usgreghash2spot  26596