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Theorem elwspths2spth 29834

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 29783	. . 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 29827	. . . 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 1117	. . 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 3208	. 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 3278	. . . 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 29725	. . . . . . 7 ⊢ (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WSPathsNOn 𝐺)𝑐) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ∧ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑐)⟨“𝑎𝑏𝑐”⟩))
9		ancom 459	. . . . . . . . 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 1945	. . . . . . . . 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 277	. . . . . . . 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 483	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉)
13		simpr 483	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → 𝑐 ∈ 𝑉)
14	12, 13	anim12i 611	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))
15		vex 3467	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
16		s3cli 14864	. . . . . . . . . . . . . 14 ⊢ ⟨“𝑎𝑏𝑐”⟩ ∈ Word V
17	15, 16	pm3.2i 469	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ V ∧ ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)
18	1	isspthonpth 29619	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 29724	. . . . . . . . . . . . 13 ⊢ (⟨“𝑎𝑏𝑐”⟩ ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ (⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐))
21		2nn0 12519	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ₀
22		iswwlksn 29705	. . . . . . . . . . . . . . 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 1436	. . . . . . . . . . . . 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 282	. . . . . . . . . . . 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 630	. . . . . . . . . . 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 479	. . . . . . . . . 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 1215	. . . . . . . . . . . . . . . 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 29598	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩ → 𝑓(Walks‘𝐺)⟨“𝑎𝑏𝑐”⟩)
31		wlklenvm1 29492	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓(Walks‘𝐺)⟨“𝑎𝑏𝑐”⟩ → (♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1))
32		simpl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1))
33		oveq1 7424	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = ((2 + 1) − 1))
34		2cn 12317	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 2 ∈ ℂ
35		pncan1 11668	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (2 ∈ ℂ → ((2 + 1) − 1) = 2)
36	34, 35	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((2 + 1) − 1) = 2
37	33, 36	eqtrdi 2781	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
38	37	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
39	38	3ad2ant1 1130	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
40	39	adantl 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) = 2)
41	32, 40	eqtrd 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((♯‘𝑓) = ((♯‘⟨“𝑎𝑏𝑐”⟩) − 1) ∧ ((⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ∧ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)) ∧ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎 ∧ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)) → (♯‘𝑓) = 2)
42	41	ex 411	. . . . . . . . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 14874	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎)
48	47	elv 3469	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎
49	48	eqcomi 2734	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0)
50		s3fv1 14875	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘1) = 𝑏)
51	50	elv 3469	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨“𝑎𝑏𝑐”⟩‘1) = 𝑏
52	51	eqcomi 2734	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1)
53		s3fv2 14876	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)
54	53	elv 3469	. . . . . . . . . . . . . . . . . . 19 ⊢ (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐
55	54	eqcomi 2734	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)
56	49, 52, 55	3pm3.2i 1336	. . . . . . . . . . . . . . . . 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 1125	. . . . . . . . . . . . . . 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 5152	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑓(SPaths‘𝐺)𝑝 ↔ 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
60		fveq1 6893	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘0) = (⟨“𝑎𝑏𝑐”⟩‘0))
61	60	eqeq2d 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑎 = (𝑝‘0) ↔ 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0)))
62		fveq1 6893	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘1) = (⟨“𝑎𝑏𝑐”⟩‘1))
63	62	eqeq2d 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑏 = (𝑝‘1) ↔ 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1)))
64		fveq1 6893	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘2) = (⟨“𝑎𝑏𝑐”⟩‘2))
65	64	eqeq2d 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑐 = (𝑝‘2) ↔ 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2)))
66	61, 63, 65	3anbi123d 1432	. . . . . . . . . . . . . . . . 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 1434	. . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . 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 411	. . . . . . . . . . . . 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 3580	. . . . . . . . . . . 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 29598	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓(SPaths‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝)
73		wlklenvp1 29488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓(Walks‘𝐺)𝑝 → (♯‘𝑝) = ((♯‘𝑓) + 1))
74		oveq1 7424	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = (2 + 1))
75		2p1e3 12384	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2 + 1) = 3
76	74, 75	eqtrdi 2781	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝑓) = 2 → ((♯‘𝑓) + 1) = 3)
77	76	eqeq2d 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑓) = 2 → ((♯‘𝑝) = ((♯‘𝑓) + 1) ↔ (♯‘𝑝) = 3))
78	77	biimpcd 248	. . . . . . . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2) → (♯‘𝑝) = 3)
81	80	3adant3 1129	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (♯‘𝑝) = 3)
82	81	adantl 480	. . . . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (𝑝‘0) ↔ (𝑝‘0) = 𝑎)
84		eqcom 2732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = (𝑝‘1) ↔ (𝑝‘1) = 𝑏)
85		eqcom 2732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = (𝑝‘2) ↔ (𝑝‘2) = 𝑐)
86	83, 84, 85	3anbi123i 1152	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) ↔ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
87	86	biimpi 215	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) → ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
88	87	3ad2ant3 1132	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 510	. . . . . . . . . . . . . . . 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 29487	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓(Walks‘𝐺)𝑝 → 𝑝 ∈ Word 𝑉)
92	72, 91	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓(SPaths‘𝐺)𝑝 → 𝑝 ∈ Word 𝑉)
93	92	3ad2ant1 1130	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → 𝑝 ∈ Word 𝑉)
94	12	anim1i 613	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
95		3anass 1092	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ↔ (𝑎 ∈ 𝑉 ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)))
96	94, 95	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) ∧ (𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉)) → (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉))
97		eqwrds3 14944	. . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . 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 248	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓(SPaths‘𝐺)𝑝 → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑓(SPaths‘𝐺)⟨“𝑎𝑏𝑐”⟩))
101	100	3ad2ant1 1130	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑓) = 2 → (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = (⟨“𝑎𝑏𝑐”⟩‘2))
106	105, 54	eqtrdi 2781	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑓) = 2 → (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)
107	106	3ad2ant2 1131	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (⟨“𝑎𝑏𝑐”⟩‘(♯‘𝑓)) = 𝑐)
108	107	ad2antlr 725	. . . . . . . . . . . . . . . . 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 1125	. . . . . . . . . . . . . . . 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 29734	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐺 ∈ UPGraph → (𝑓(Walks‘𝐺)𝑝 → 𝑝 ∈ (WWalks‘𝐺)))
111	110	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑎 ∈ 𝑉) → (𝑓(Walks‘𝐺)𝑝 → 𝑝 ∈ (WWalks‘𝐺)))
112	111	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝 ∈ (WWalks‘𝐺) ↔ ⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺)))
118	117	bicomd 222	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ↔ 𝑝 ∈ (WWalks‘𝐺)))
119	118	adantl 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”⟩) → (⟨“𝑎𝑏𝑐”⟩ ∈ (WWalks‘𝐺) ↔ 𝑝 ∈ (WWalks‘𝐺)))
120	116, 119	mpbird 256	. . . . . . . . . . . . . . . . . 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 14877	. . . . . . . . . . . . . . . . . . 19 ⊢ (♯‘⟨“𝑎𝑏𝑐”⟩) = 3
122		df-3 12306	. . . . . . . . . . . . . . . . . . 19 ⊢ 3 = (2 + 1)
123	121, 122	eqtri 2753	. . . . . . . . . . . . . . . . . 18 ⊢ (♯‘⟨“𝑎𝑏𝑐”⟩) = (2 + 1)
124	120, 123	jctir 519	. . . . . . . . . . . . . . . . 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 1125	. . . . . . . . . . . . . . . 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 510	. . . . . . . . . . . . . . 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 685	. . . . . . . . . . . . . 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 411	. . . . . . . . . . . . 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 1928	. . . . . . . . . . . 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 211	. . . . . . . . . . 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 479	. . . . . . . . . 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 278	. . . . . . . . 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 1916	. . . . . . . 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 282	. . . . . . 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 282	. . . . . 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 577	. . . . 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 3208	. . . 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 282	. . 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 3167	. 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 304	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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∃wrex 3060 Vcvv 3463 class class class wbr 5148 ‘cfv 6547 (class class class)co 7417 ℂcc 11136 0cc0 11138 1c1 11139 + caddc 11141 − cmin 11474 2c2 12297 3c3 12298 ℕ₀cn0 12502 ♯chash 14321 Word cword 14496 ⟨“cs3 14825 Vtxcvtx 28865 UPGraphcupgr 28949 Walkscwlks 29466 SPathscspths 29583 SPathsOncspthson 29585 WWalkscwwlks 29692 WWalksN cwwlksn 29693 WWalksNOn cwwlksnon 29694 WSPathsN cwwspthsn 29695 WSPathsNOn cwwspthsnon 29696

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-ac2 10486 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7870 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-er 8723 df-map 8845 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-dju 9924 df-card 9962 df-ac 10139 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-xnn0 12575 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-hash 14322 df-word 14497 df-concat 14553 df-s1 14578 df-s2 14831 df-s3 14832 df-edg 28917 df-uhgr 28927 df-upgr 28951 df-wlks 29469 df-wlkson 29470 df-trls 29562 df-trlson 29563 df-pths 29586 df-spths 29587 df-spthson 29589 df-wwlks 29697 df-wwlksn 29698 df-wwlksnon 29699 df-wspthsn 29700 df-wspthsnon 29701

This theorem is referenced by: (None)

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