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Theorem elwspths2spth 29992

Description: A simple path of length 2 between two vertices as length 3 string in a pseudograph. (Contributed by Alexander van der Vekens, 28-Feb-2018.) (Revised by AV, 18-May-2021.) (Proof shortened by AV, 16-Mar-2022.)

**Hypothesis**
Ref	Expression
elwwlks2.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
elwspths2spth	⊢ (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))

Distinct variable groups: 𝐺,𝑎,𝑏,𝑐,𝑓,𝑝 𝑉,𝑎,𝑏,𝑐,𝑓,𝑝 𝑊,𝑎,𝑏,𝑐,𝑓,𝑝

**Proof of Theorem elwspths2spth**
Step	Hyp	Ref	Expression
1		elwwlks2.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	wspthsnwspthsnon 29938	. . 3 ⊢ (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐))
3	2	a1i 11	. 2 ⊢ (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)))
4	1	elwspths2on 29984	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐))))
5	4	3expb 1120	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐))))
6	5	2rexbidva 3197	. 2 ⊢ (𝐺 ∈ UPGraph → (∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ ∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐))))
7		rexcom 3263	. . . 4 ⊢ (∃𝑐 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)))
8		wspthnon 29880	. . . . . . 7 ⊢ (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩))
9		ancom 460	. . . . . . . . 9 ⊢ ((⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩) ↔ (∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)))
10		19.41v 1950	. . . . . . . . 9 ⊢ (∃𝑓(𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)) ↔ (∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)))
11	9, 10	bitr4i 278	. . . . . . . 8 ⊢ ((⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩) ↔ ∃𝑓(𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)))
12		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉)
13		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → 𝑐 ∈ 𝑉)
14	12, 13	anim12i 613	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))
15		vex 3442	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
16		s3cli 14802	. . . . . . . . . . . . . 14 ⊢ ⟨“𝑎𝑏𝑐”⟩ ∈ Word V
17	15, 16	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ V ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)
18	1	isspthonpth 29771	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ (𝑓 ∈ V ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)) → (𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ↔ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)))
19	14, 17, 18	sylancl 586	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ↔ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)))
20		wwlknon 29879	. . . . . . . . . . . . 13 ⊢ (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))
21		2nn0 12416	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ₀
22		iswwlksn 29860	. . . . . . . . . . . . . . 15 ⊢ (2 ∈ ℕ₀ → (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1))))
23	21, 22	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1))))
24	23	3anbi1d 1442	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → ((⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) ↔ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)))
25	20, 24	bitrid 283	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)))
26	19, 25	anbi12d 632	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → ((𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)) ↔ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))))
27	26	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)) ↔ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))))
28	16	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)
29		simprl1 1219	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩)
30		spthiswlk 29748	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ → 𝑓(Walks‘𝐺)⟨“𝑎𝑏𝑐”⟩)
31		wlklenvm1 29644	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓(Walks‘𝐺)⟨“𝑎𝑏𝑐”⟩ → (♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1))
32		simpl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1))
33		oveq1 7363	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = ((2 + 1) − 1))
34		2cn 12218	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 2 ∈ ℂ
35		pncan1 11559	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (2 ∈ ℂ → ((2 + 1) − 1) = 2)
36	34, 35	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((2 + 1) − 1) = 2
37	33, 36	eqtrdi 2785	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
38	37	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
39	38	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
40	39	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
41	32, 40	eqtrd 2769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (♯‘𝑓) = 2)
42	41	ex 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) → (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → (♯‘𝑓) = 2))
43	30, 31, 42	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ → (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → (♯‘𝑓) = 2))
44	43	3ad2ant1 1133	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) → (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → (♯‘𝑓) = 2))
45	44	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (♯‘𝑓) = 2)
46	45	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → (♯‘𝑓) = 2)
47		s3fv0 14812	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎)
48	47	elv 3443	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎
49	48	eqcomi 2743	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0)
50		s3fv1 14813	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘1) = 𝑏)
51	50	elv 3443	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨“𝑎𝑏𝑐”⟩‘1) = 𝑏
52	51	eqcomi 2743	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1)
53		s3fv2 14814	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)
54	53	elv 3443	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐
55	54	eqcomi 2743	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)
56	49, 52, 55	3pm3.2i 1340	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))
57	56	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)))
58	29, 46, 57	3jca 1128	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))))
59		breq2 5100	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑓(SPaths‘𝐺)𝑝 ↔ 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
60		fveq1 6831	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘0) = (⟨“𝑎𝑏𝑐”⟩‘0))
61	60	eqeq2d 2745	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑎 = (𝑝‘0) ↔ 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0)))
62		fveq1 6831	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘1) = (⟨“𝑎𝑏𝑐”⟩‘1))
63	62	eqeq2d 2745	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑏 = (𝑝‘1) ↔ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1)))
64		fveq1 6831	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘2) = (⟨“𝑎𝑏𝑐”⟩‘2))
65	64	eqeq2d 2745	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑐 = (𝑝‘2) ↔ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)))
66	61, 63, 65	3anbi123d 1438	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) ↔ (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))))
67	59, 66	3anbi13d 1440	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) ↔ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)))))
68	67	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) ↔ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0) ∧ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1) ∧ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)))))
69	58, 68	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))) → (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))
70	69	ex 412	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
71	28, 70	spcimedv 3547	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → ∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
72		spthiswlk 29748	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓(SPaths‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝)
73		wlklenvp1 29641	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓(Walks‘𝐺)𝑝 → (♯‘𝑝) = ((♯‘𝑓) + 1))
74		oveq1 7363	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = (2 + 1))
75		2p1e3 12280	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2 + 1) = 3
76	74, 75	eqtrdi 2785	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = 3)
77	76	eqeq2d 2745	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑓) = 2 → ((♯‘𝑝) = ((♯‘𝑓) + 1) ↔ (♯‘𝑝) = 3))
78	77	biimpcd 249	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑝) = ((♯‘𝑓) + 1) → ((♯‘𝑓) = 2 → (♯‘𝑝) = 3))
79	72, 73, 78	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓(SPaths‘𝐺)𝑝 → ((♯‘𝑓) = 2 → (♯‘𝑝) = 3))
80	79	imp 406	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2) → (♯‘𝑝) = 3)
81	80	3adant3 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (♯‘𝑝) = 3)
82	81	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → (♯‘𝑝) = 3)
83		eqcom 2741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (𝑝‘0) ↔ (𝑝‘0) = 𝑎)
84		eqcom 2741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = (𝑝‘1) ↔ (𝑝‘1) = 𝑏)
85		eqcom 2741	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = (𝑝‘2) ↔ (𝑝‘2) = 𝑐)
86	83, 84, 85	3anbi123i 1155	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) ↔ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
87	86	biimpi 216	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) → ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
88	87	3ad2ant3 1135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
89	88	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
90	82, 89	jca 511	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → ((♯‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐)))
91	1	wlkpwrd 29640	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓(Walks‘𝐺)𝑝 → 𝑝 ∈ Word 𝑉)
92	72, 91	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓(SPaths‘𝐺)𝑝 → 𝑝 ∈ Word 𝑉)
93	92	3ad2ant1 1133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → 𝑝 ∈ Word 𝑉)
94	12	anim1i 615	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
95		3anass 1094	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ↔ (𝑎 ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
96	94, 95	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))
97		eqwrds3 14882	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑝 ∈ Word 𝑉 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((♯‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))))
98	93, 96, 97	syl2anr 597	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((♯‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))))
99	90, 98	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → 𝑝 = ⟨“𝑎𝑏𝑐”⟩)
100	59	biimpcd 249	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓(SPaths‘𝐺)𝑝 → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
101	100	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
102	101	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
103	102	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩)
104	48	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎)
105		fveq2 6832	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑓) = 2 → (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = (⟨“𝑎𝑏𝑐”⟩‘2))
106	105, 54	eqtrdi 2785	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑓) = 2 → (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)
107	106	3ad2ant2 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)
108	107	ad2antlr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)
109	103, 104, 108	3jca 1128	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐))
110		wlkiswwlks1 29889	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐺 ∈ UPGraph → (𝑓(Walks‘𝐺)𝑝 → 𝑝 ∈ (WWalks‘𝐺)))
111	110	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) → (𝑓(Walks‘𝐺)𝑝 → 𝑝 ∈ (WWalks‘𝐺)))
112	111	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑓(Walks‘𝐺)𝑝 → 𝑝 ∈ (WWalks‘𝐺)))
113	72, 112	syl5com 31	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓(SPaths‘𝐺)𝑝 → (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑝 ∈ (WWalks‘𝐺)))
114	113	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → 𝑝 ∈ (WWalks‘𝐺)))
115	114	impcom 407	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → 𝑝 ∈ (WWalks‘𝐺))
116	115	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑝 ∈ (WWalks‘𝐺))
117		eleq1 2822	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝 ∈ (WWalks‘𝐺) ↔ ⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺)))
118	117	bicomd 223	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ↔ 𝑝 ∈ (WWalks‘𝐺)))
119	118	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ↔ 𝑝 ∈ (WWalks‘𝐺)))
120	116, 119	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → ⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺))
121		s3len 14815	. . . . . . . . . . . . . . . . . . 19 ⊢ (♯‘⟨“𝑎𝑏𝑐”⟩) = 3
122		df-3 12207	. . . . . . . . . . . . . . . . . . 19 ⊢ 3 = (2 + 1)
123	121, 122	eqtri 2757	. . . . . . . . . . . . . . . . . 18 ⊢ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)
124	120, 123	jctir 520	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)))
125	54	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)
126	124, 104, 125	3jca 1128	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))
127	109, 126	jca 511	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)))
128	99, 127	mpdan 687	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ (𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))) → ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)))
129	128	ex 412	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))))
130	129	exlimdv 1934	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → ((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))))
131	71, 130	impbid 212	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) ↔ ∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
132	131	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (((𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) ↔ ∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
133	27, 132	bitrd 279	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)) ↔ ∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
134	133	exbidv 1922	. . . . . . . 8 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (∃𝑓(𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)) ↔ ∃𝑓∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
135	11, 134	bitrid 283	. . . . . . 7 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩) ↔ ∃𝑓∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
136	8, 135	bitrid 283	. . . . . 6 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ ∃𝑓∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
137	136	pm5.32da 579	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → ((𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
138	137	2rexbidva 3197	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) → (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
139	7, 138	bitrid 283	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) → (∃𝑐 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
140	139	rexbidva 3156	. 2 ⊢ (𝐺 ∈ UPGraph → (∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐)) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
141	3, 6, 140	3bitrd 305	1 ⊢ (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∃wex 1780 ∈ wcel 2113 ∃wrex 3058 Vcvv 3438 class class class wbr 5096 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 0cc0 11024 1c1 11025 + caddc 11027 − cmin 11362 2c2 12198 3c3 12199 ℕ₀cn0 12399 ♯chash 14251 Word cword 14434 ⟨“cs3 14763 Vtxcvtx 29018 UPGraphcupgr 29102 Walkscwlks 29619 SPathscspths 29733 SPathsOncspthson 29735 WWalkscwwlks 29847 WWalksN cwwlksn 29848 WWalksNOn cwwlksnon 29849 WSPathsN cwwspthsn 29850 WSPathsNOn cwwspthsnon 29851

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-ac2 10371 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8633 df-map 8763 df-pm 8764 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-dju 9811 df-card 9849 df-ac 10024 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-n0 12400 df-xnn0 12473 df-z 12487 df-uz 12750 df-fz 13422 df-fzo 13569 df-hash 14252 df-word 14435 df-concat 14492 df-s1 14518 df-s2 14769 df-s3 14770 df-edg 29070 df-uhgr 29080 df-upgr 29104 df-wlks 29622 df-wlkson 29623 df-trls 29713 df-trlson 29714 df-pths 29736 df-spths 29737 df-spthson 29739 df-wwlks 29852 df-wwlksn 29853 df-wwlksnon 29854 df-wspthsn 29855 df-wspthsnon 29856

This theorem is referenced by: (None)

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