Theorem el2wlkonotot0 26399

Description: A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)

Assertion

Ref	Expression
el2wlkonotot0	⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑅(𝑉 2WalksOnOt 𝐸)𝑆) ↔ (𝐴 = 𝑅 ∧ 𝐶 = 𝑆 ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))

Distinct variable groups:   𝐴,𝑓,𝑝   𝐵,𝑓,𝑝   𝐶,𝑓,𝑝   𝑓,𝐸,𝑝   𝑓,𝑉,𝑝   𝑅,𝑓,𝑝   𝑆,𝑓,𝑝   𝑓,𝑋,𝑝   𝑓,𝑌,𝑝

Proof of Theorem el2wlkonotot0

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		el2wlkonot 26396	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑅(𝑉 2WalksOnOt 𝐸)𝑆) ↔ ∃𝑏 ∈ 𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))))
2		19.42vv 1907	. . . . . 6 ⊢ (∃𝑓∃𝑝(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
3	2	bicomi 213	. . . . 5 ⊢ ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑓∃𝑝(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
4	3	rexbii 3023	. . . 4 ⊢ (∃𝑏 ∈ 𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑏 ∈ 𝑉 ∃𝑓∃𝑝(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
5		rexcom4 3198	. . . 4 ⊢ (∃𝑏 ∈ 𝑉 ∃𝑓∃𝑝(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑓∃𝑏 ∈ 𝑉 ∃𝑝(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
6		rexcom4 3198	. . . . 5 ⊢ (∃𝑏 ∈ 𝑉 ∃𝑝(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑝∃𝑏 ∈ 𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
7	6	exbii 1764	. . . 4 ⊢ (∃𝑓∃𝑏 ∈ 𝑉 ∃𝑝(⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑓∃𝑝∃𝑏 ∈ 𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
8	4, 5, 7	3bitri 285	. . 3 ⊢ (∃𝑏 ∈ 𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑓∃𝑝∃𝑏 ∈ 𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
9	8	a1i 11	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) → (∃𝑏 ∈ 𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑓∃𝑝∃𝑏 ∈ 𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))))
10		eqcom 2617	. . . . . . . . . . 11 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ↔ ⟨𝑅, 𝑏, 𝑆⟩ = ⟨𝐴, 𝐵, 𝐶⟩)
11		df-ot 4134	. . . . . . . . . . . 12 ⊢ ⟨𝑅, 𝑏, 𝑆⟩ = ⟨⟨𝑅, 𝑏⟩, 𝑆⟩
12		df-ot 4134	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
13	11, 12	eqeq12i 2624	. . . . . . . . . . 11 ⊢ (⟨𝑅, 𝑏, 𝑆⟩ = ⟨𝐴, 𝐵, 𝐶⟩ ↔ ⟨⟨𝑅, 𝑏⟩, 𝑆⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
14	10, 13	bitri 263	. . . . . . . . . 10 ⊢ (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ↔ ⟨⟨𝑅, 𝑏⟩, 𝑆⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
15	14	a1i 11	. . . . . . . . 9 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ↔ ⟨⟨𝑅, 𝑏⟩, 𝑆⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩))
16		opex 4859	. . . . . . . . . . . . . 14 ⊢ ⟨𝑅, 𝑏⟩ ∈ V
17	16	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉) → ⟨𝑅, 𝑏⟩ ∈ V)
18		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉) → 𝑆 ∈ 𝑉)
19	17, 18	jca 553	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉) → (⟨𝑅, 𝑏⟩ ∈ V ∧ 𝑆 ∈ 𝑉))
20	19	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) → (⟨𝑅, 𝑏⟩ ∈ V ∧ 𝑆 ∈ 𝑉))
21	20	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (⟨𝑅, 𝑏⟩ ∈ V ∧ 𝑆 ∈ 𝑉))
22		opthg 4872	. . . . . . . . . 10 ⊢ ((⟨𝑅, 𝑏⟩ ∈ V ∧ 𝑆 ∈ 𝑉) → (⟨⟨𝑅, 𝑏⟩, 𝑆⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ↔ (⟨𝑅, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ 𝑆 = 𝐶)))
23	21, 22	syl 17	. . . . . . . . 9 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (⟨⟨𝑅, 𝑏⟩, 𝑆⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ↔ (⟨𝑅, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ 𝑆 = 𝐶)))
24		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉) → 𝑅 ∈ 𝑉)
25	24	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) → 𝑅 ∈ 𝑉)
26		opthg 4872	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (⟨𝑅, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑅 = 𝐴 ∧ 𝑏 = 𝐵)))
27	25, 26	sylan 487	. . . . . . . . . 10 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (⟨𝑅, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑅 = 𝐴 ∧ 𝑏 = 𝐵)))
28	27	anbi1d 737	. . . . . . . . 9 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((⟨𝑅, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ∧ 𝑆 = 𝐶) ↔ ((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶)))
29	15, 23, 28	3bitrd 293	. . . . . . . 8 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ↔ ((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶)))
30		eqcom 2617	. . . . . . . . . . . . . 14 ⊢ (𝑅 = 𝐴 ↔ 𝐴 = 𝑅)
31	30	biimpi 205	. . . . . . . . . . . . 13 ⊢ (𝑅 = 𝐴 → 𝐴 = 𝑅)
32	31	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) → 𝐴 = 𝑅)
33		eqcom 2617	. . . . . . . . . . . . 13 ⊢ (𝑆 = 𝐶 ↔ 𝐶 = 𝑆)
34	33	biimpi 205	. . . . . . . . . . . 12 ⊢ (𝑆 = 𝐶 → 𝐶 = 𝑆)
35	32, 34	anim12i 588	. . . . . . . . . . 11 ⊢ (((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → (𝐴 = 𝑅 ∧ 𝐶 = 𝑆))
36	35	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → (𝐴 = 𝑅 ∧ 𝐶 = 𝑆))
37		simpr1 1060	. . . . . . . . . . 11 ⊢ ((((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → 𝑓(𝑉 Walks 𝐸)𝑝)
38		simpr2 1061	. . . . . . . . . . 11 ⊢ ((((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → (#‘𝑓) = 2)
39		eqtr2 2630	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 = 𝐴 ∧ 𝑅 = (𝑝‘0)) → 𝐴 = (𝑝‘0))
40	39	ex 449	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 = 𝐴 → (𝑅 = (𝑝‘0) → 𝐴 = (𝑝‘0)))
41	40	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → (𝑅 = (𝑝‘0) → 𝐴 = (𝑝‘0)))
42		eqtr2 2630	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 = 𝐵 ∧ 𝑏 = (𝑝‘1)) → 𝐵 = (𝑝‘1))
43	42	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝐵 → (𝑏 = (𝑝‘1) → 𝐵 = (𝑝‘1)))
44	43	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑏 = (𝑝‘1) → 𝐵 = (𝑝‘1)))
45	44	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → (𝑏 = (𝑝‘1) → 𝐵 = (𝑝‘1)))
46		eqtr2 2630	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 = 𝐶 ∧ 𝑆 = (𝑝‘2)) → 𝐶 = (𝑝‘2))
47	46	ex 449	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 = 𝐶 → (𝑆 = (𝑝‘2) → 𝐶 = (𝑝‘2)))
48	47	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → (𝑆 = (𝑝‘2) → 𝐶 = (𝑝‘2)))
49	41, 45, 48	3anim123d 1398	. . . . . . . . . . . . . 14 ⊢ (((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → ((𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)) → (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
50	49	com12 32	. . . . . . . . . . . . 13 ⊢ ((𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)) → (((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
51	50	3ad2ant3 1077	. . . . . . . . . . . 12 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))) → (((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
52	51	impcom 445	. . . . . . . . . . 11 ⊢ ((((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))
53	37, 38, 52	3jca 1235	. . . . . . . . . 10 ⊢ ((((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
54	36, 53	jca 553	. . . . . . . . 9 ⊢ ((((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
55	54	ex 449	. . . . . . . 8 ⊢ (((𝑅 = 𝐴 ∧ 𝑏 = 𝐵) ∧ 𝑆 = 𝐶) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))) → ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
56	29, 55	syl6bi 242	. . . . . . 7 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))) → ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))
57	56	impd 446	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) ∧ 𝑏 ∈ 𝑉) → ((⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
58	57	rexlimdva 3013	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) → (∃𝑏 ∈ 𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) → ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
59		el2wlkonotlem 26389	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → (𝑝‘1) ∈ 𝑉)
60	59	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) ∧ 𝐵 = (𝑝‘1)) → (𝑝‘1) ∈ 𝑉)
61		eleq1 2676	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = (𝑝‘1) → (𝐵 ∈ 𝑉 ↔ (𝑝‘1) ∈ 𝑉))
62	61	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) ∧ 𝐵 = (𝑝‘1)) → (𝐵 ∈ 𝑉 ↔ (𝑝‘1) ∈ 𝑉))
63	60, 62	mpbird 246	. . . . . . . . . . . . . . 15 ⊢ (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) ∧ 𝐵 = (𝑝‘1)) → 𝐵 ∈ 𝑉)
64	63	a1d 25	. . . . . . . . . . . . . 14 ⊢ (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) ∧ 𝐵 = (𝑝‘1)) → ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) → 𝐵 ∈ 𝑉))
65	64	ex 449	. . . . . . . . . . . . 13 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → (𝐵 = (𝑝‘1) → ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) → 𝐵 ∈ 𝑉)))
66	65	ex 449	. . . . . . . . . . . 12 ⊢ (𝑓(𝑉 Walks 𝐸)𝑝 → ((#‘𝑓) = 2 → (𝐵 = (𝑝‘1) → ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) → 𝐵 ∈ 𝑉))))
67	66	com13 86	. . . . . . . . . . 11 ⊢ (𝐵 = (𝑝‘1) → ((#‘𝑓) = 2 → (𝑓(𝑉 Walks 𝐸)𝑝 → ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) → 𝐵 ∈ 𝑉))))
68	67	3ad2ant2 1076	. . . . . . . . . 10 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((#‘𝑓) = 2 → (𝑓(𝑉 Walks 𝐸)𝑝 → ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) → 𝐵 ∈ 𝑉))))
69	68	com13 86	. . . . . . . . 9 ⊢ (𝑓(𝑉 Walks 𝐸)𝑝 → ((#‘𝑓) = 2 → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) → 𝐵 ∈ 𝑉))))
70	69	3imp 1249	. . . . . . . 8 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) → 𝐵 ∈ 𝑉))
71	70	impcom 445	. . . . . . 7 ⊢ (((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝐵 ∈ 𝑉)
72		simpl 472	. . . . . . . . . . 11 ⊢ ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) → 𝐴 = 𝑅)
73	72	ad2antrr 758	. . . . . . . . . 10 ⊢ ((((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → 𝐴 = 𝑅)
74		eqcom 2617	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 ↔ 𝐵 = 𝑏)
75	74	biimpi 205	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → 𝐵 = 𝑏)
76	75	adantl 481	. . . . . . . . . 10 ⊢ ((((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → 𝐵 = 𝑏)
77		simpr 476	. . . . . . . . . . 11 ⊢ ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) → 𝐶 = 𝑆)
78	77	ad2antrr 758	. . . . . . . . . 10 ⊢ ((((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → 𝐶 = 𝑆)
79	73, 76, 78	oteq123d 4355	. . . . . . . . 9 ⊢ ((((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → ⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩)
80		simpr1 1060	. . . . . . . . . . 11 ⊢ (((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑓(𝑉 Walks 𝐸)𝑝)
81	80	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → 𝑓(𝑉 Walks 𝐸)𝑝)
82		simplr2 1097	. . . . . . . . . 10 ⊢ ((((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → (#‘𝑓) = 2)
83		eqtr2 2630	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 = 𝑅 ∧ 𝐴 = (𝑝‘0)) → 𝑅 = (𝑝‘0))
84	83	ex 449	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 = 𝑅 → (𝐴 = (𝑝‘0) → 𝑅 = (𝑝‘0)))
85	84	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) → (𝐴 = (𝑝‘0) → 𝑅 = (𝑝‘0)))
86	85	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 𝐵 ∧ (𝐴 = 𝑅 ∧ 𝐶 = 𝑆)) → (𝐴 = (𝑝‘0) → 𝑅 = (𝑝‘0)))
87		eqtr2 2630	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 = 𝑏 ∧ 𝐵 = (𝑝‘1)) → 𝑏 = (𝑝‘1))
88	87	ex 449	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 = 𝑏 → (𝐵 = (𝑝‘1) → 𝑏 = (𝑝‘1)))
89	88	eqcoms 2618	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝐵 → (𝐵 = (𝑝‘1) → 𝑏 = (𝑝‘1)))
90	89	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 𝐵 ∧ (𝐴 = 𝑅 ∧ 𝐶 = 𝑆)) → (𝐵 = (𝑝‘1) → 𝑏 = (𝑝‘1)))
91		eqtr2 2630	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 = 𝑆 ∧ 𝐶 = (𝑝‘2)) → 𝑆 = (𝑝‘2))
92	91	ex 449	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 = 𝑆 → (𝐶 = (𝑝‘2) → 𝑆 = (𝑝‘2)))
93	92	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) → (𝐶 = (𝑝‘2) → 𝑆 = (𝑝‘2)))
94	93	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 = 𝐵 ∧ (𝐴 = 𝑅 ∧ 𝐶 = 𝑆)) → (𝐶 = (𝑝‘2) → 𝑆 = (𝑝‘2)))
95	86, 90, 94	3anim123d 1398	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 = 𝐵 ∧ (𝐴 = 𝑅 ∧ 𝐶 = 𝑆)) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))
96	95	ex 449	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝐵 → ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) → ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
97	96	com13 86	. . . . . . . . . . . . 13 ⊢ ((𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) → (𝑏 = 𝐵 → (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
98	97	3ad2ant3 1077	. . . . . . . . . . . 12 ⊢ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) → (𝑏 = 𝐵 → (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
99	98	impcom 445	. . . . . . . . . . 11 ⊢ (((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑏 = 𝐵 → (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))
100	99	imp 444	. . . . . . . . . 10 ⊢ ((((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))
101	81, 82, 100	3jca 1235	. . . . . . . . 9 ⊢ ((((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))
102	79, 101	jca 553	. . . . . . . 8 ⊢ ((((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))))
103	102	a1d 25	. . . . . . 7 ⊢ ((((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ∧ 𝑏 = 𝐵) → (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))))
104	71, 103	rspcimedv 3284	. . . . . 6 ⊢ (((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))))
105	104	com12 32	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) → (((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → ∃𝑏 ∈ 𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2))))))
106	58, 105	impbid 201	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) → (∃𝑏 ∈ 𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
107	106	2exbidv 1839	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) → (∃𝑓∃𝑝∃𝑏 ∈ 𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ ∃𝑓∃𝑝((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
108		df-3an 1033	. . . 4 ⊢ ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆 ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
109		19.42vv 1907	. . . . 5 ⊢ (∃𝑓∃𝑝((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
110	109	bicomi 213	. . . 4 ⊢ (((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑓∃𝑝((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
111	108, 110	bitri 263	. . 3 ⊢ ((𝐴 = 𝑅 ∧ 𝐶 = 𝑆 ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑓∃𝑝((𝐴 = 𝑅 ∧ 𝐶 = 𝑆) ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
112	107, 111	syl6bbr 277	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) → (∃𝑓∃𝑝∃𝑏 ∈ 𝑉 (⟨𝐴, 𝐵, 𝐶⟩ = ⟨𝑅, 𝑏, 𝑆⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑅 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑆 = (𝑝‘2)))) ↔ (𝐴 = 𝑅 ∧ 𝐶 = 𝑆 ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
113	1, 9, 112	3bitrd 293	1 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑅(𝑉 2WalksOnOt 𝐸)𝑆) ↔ (𝐴 = 𝑅 ∧ 𝐶 = 𝑆 ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173  ⟨cop 4131  ⟨cotp 4133   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  #chash 12979   Walks cwalk 26026   2WalksOnOt c2wlkonot 26382

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-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-wlk 26036  df-wlkon 26042  df-2wlkonot 26385

This theorem is referenced by:  el2wlkonotot  26400  el2wlkonotot1  26401  el2wlksotot  26409