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Theorem el2wlkonot 26396
 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
el2wlkonot (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
Distinct variable groups:   𝐴,𝑏,𝑓,𝑝   𝐶,𝑏,𝑓,𝑝   𝐸,𝑏,𝑓,𝑝   𝑇,𝑏,𝑓,𝑝   𝑉,𝑏,𝑓,𝑝   𝑋,𝑏,𝑓,𝑝   𝑌,𝑏,𝑓,𝑝

Proof of Theorem el2wlkonot
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 2wlkonot 26392 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐶))})
21eleq2d 2673 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ 𝑇 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐶))}))
3 fveq2 6103 . . . . . . . . 9 (𝑡 = 𝑇 → (1st𝑡) = (1st𝑇))
43fveq2d 6107 . . . . . . . 8 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
54eqeq1d 2612 . . . . . . 7 (𝑡 = 𝑇 → ((1st ‘(1st𝑡)) = 𝐴 ↔ (1st ‘(1st𝑇)) = 𝐴))
63fveq2d 6107 . . . . . . . 8 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
76eqeq1d 2612 . . . . . . 7 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) = (𝑝‘1) ↔ (2nd ‘(1st𝑇)) = (𝑝‘1)))
8 fveq2 6103 . . . . . . . 8 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
98eqeq1d 2612 . . . . . . 7 (𝑡 = 𝑇 → ((2nd𝑡) = 𝐶 ↔ (2nd𝑇) = 𝐶))
105, 7, 93anbi123d 1391 . . . . . 6 (𝑡 = 𝑇 → (((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐶) ↔ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)))
11103anbi3d 1397 . . . . 5 (𝑡 = 𝑇 → ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐶)) ↔ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
12112exbidv 1839 . . . 4 (𝑡 = 𝑇 → (∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐶)) ↔ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
1312elrab 3331 . . 3 (𝑇 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐶))} ↔ (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
142, 13syl6bb 275 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)))))
15 simpl 472 . . . . . . . . 9 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑉𝑋𝐸𝑌))
16 vex 3176 . . . . . . . . . . 11 𝑓 ∈ V
17 vex 3176 . . . . . . . . . . 11 𝑝 ∈ V
1816, 17pm3.2i 470 . . . . . . . . . 10 (𝑓 ∈ V ∧ 𝑝 ∈ V)
1918a1i 11 . . . . . . . . 9 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑓 ∈ V ∧ 𝑝 ∈ V))
20 simpr 476 . . . . . . . . 9 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝐴𝑉𝐶𝑉))
21 iswlkon 26062 . . . . . . . . 9 (((𝑉𝑋𝐸𝑌) ∧ (𝑓 ∈ V ∧ 𝑝 ∈ V) ∧ (𝐴𝑉𝐶𝑉)) → (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶)))
2215, 19, 20, 21syl3anc 1318 . . . . . . . 8 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶)))
23223anbi1d 1395 . . . . . . 7 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)) ↔ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
2423anbi2d 736 . . . . . 6 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) ↔ (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)))))
25 simpl 472 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑉𝐶𝑉) → 𝐴𝑉)
2625adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → 𝐴𝑉)
2726adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) → 𝐴𝑉)
2827adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) ∧ (#‘𝑓) = 2) → 𝐴𝑉)
29 el2wlkonotlem 26389 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → (𝑝‘1) ∈ 𝑉)
3029ex 449 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓(𝑉 Walks 𝐸)𝑝 → ((#‘𝑓) = 2 → (𝑝‘1) ∈ 𝑉))
31303ad2ant1 1075 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → ((#‘𝑓) = 2 → (𝑝‘1) ∈ 𝑉))
3231adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) → ((#‘𝑓) = 2 → (𝑝‘1) ∈ 𝑉))
3332imp 444 . . . . . . . . . . . . . . . . . . 19 ((((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) ∧ (#‘𝑓) = 2) → (𝑝‘1) ∈ 𝑉)
34 simpr 476 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑉𝐶𝑉) → 𝐶𝑉)
3534adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → 𝐶𝑉)
3635adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) → 𝐶𝑉)
3736adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) ∧ (#‘𝑓) = 2) → 𝐶𝑉)
38 el2xptp0 7103 . . . . . . . . . . . . . . . . . . 19 ((𝐴𝑉 ∧ (𝑝‘1) ∈ 𝑉𝐶𝑉) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)) ↔ 𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩))
3928, 33, 37, 38syl3anc 1318 . . . . . . . . . . . . . . . . . 18 ((((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) ∧ (#‘𝑓) = 2) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)) ↔ 𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩))
40 oteq2 4350 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑝‘1) = 𝑏 → ⟨𝐴, (𝑝‘1), 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩)
4140eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑏 = (𝑝‘1) → ⟨𝐴, (𝑝‘1), 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩)
4241eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑏 = (𝑝‘1) → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ↔ 𝑇 = ⟨𝐴, 𝑏, 𝐶⟩))
4342biimpcd 238 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → (𝑏 = (𝑝‘1) → 𝑇 = ⟨𝐴, 𝑏, 𝐶⟩))
4443adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) → (𝑏 = (𝑝‘1) → 𝑇 = ⟨𝐴, 𝑏, 𝐶⟩))
4544ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → (𝑏 = (𝑝‘1) → 𝑇 = ⟨𝐴, 𝑏, 𝐶⟩))
4645impcom 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → 𝑇 = ⟨𝐴, 𝑏, 𝐶⟩)
47 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → 𝑓(𝑉 Walks 𝐸)𝑝)
4847adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) → 𝑓(𝑉 Walks 𝐸)𝑝)
4948ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → 𝑓(𝑉 Walks 𝐸)𝑝)
5049adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → 𝑓(𝑉 Walks 𝐸)𝑝)
51 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → (#‘𝑓) = 2)
5251adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) → (#‘𝑓) = 2)
5352ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → (#‘𝑓) = 2)
5453adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → (#‘𝑓) = 2)
55 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑝‘0) = 𝐴𝐴 = (𝑝‘0))
5655biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑝‘0) = 𝐴𝐴 = (𝑝‘0))
5756adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → 𝐴 = (𝑝‘0))
5857adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → 𝐴 = (𝑝‘0))
59 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → 𝑏 = (𝑝‘1))
60 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((#‘𝑓) = 2 → (𝑝‘(#‘𝑓)) = (𝑝‘2))
6160eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((#‘𝑓) = 2 → ((𝑝‘(#‘𝑓)) = 𝐶 ↔ (𝑝‘2) = 𝐶))
62 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑝‘2) = 𝐶𝐶 = (𝑝‘2))
6362biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑝‘2) = 𝐶𝐶 = (𝑝‘2))
6461, 63syl6bi 242 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((#‘𝑓) = 2 → ((𝑝‘(#‘𝑓)) = 𝐶𝐶 = (𝑝‘2)))
6564adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝑝‘(#‘𝑓)) = 𝐶𝐶 = (𝑝‘2)))
6665adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) → ((𝑝‘(#‘𝑓)) = 𝐶𝐶 = (𝑝‘2)))
6766imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) → 𝐶 = (𝑝‘2))
6867adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → 𝐶 = (𝑝‘2))
6968adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → 𝐶 = (𝑝‘2))
7058, 59, 693jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))
7150, 54, 703jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
7246, 71jca 553 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑏 = (𝑝‘1) ∧ (((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴)) → (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
7372ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏 = (𝑝‘1) → ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
7473adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) ∧ 𝑏 = (𝑝‘1)) → ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
7529, 74rspcimedv 3284 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
7675com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2)) ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (𝑝‘0) = 𝐴) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
7776exp41 636 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝑝‘(#‘𝑓)) = 𝐶 → ((𝑝‘0) = 𝐴 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))))
7877com15 99 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝑝‘(#‘𝑓)) = 𝐶 → ((𝑝‘0) = 𝐴 → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))))
7978pm2.43i 50 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → ((𝑝‘(#‘𝑓)) = 𝐶 → ((𝑝‘0) = 𝐴 → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))))
8079ex 449 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓(𝑉 Walks 𝐸)𝑝 → ((#‘𝑓) = 2 → ((𝑝‘(#‘𝑓)) = 𝐶 → ((𝑝‘0) = 𝐴 → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))))
8180com24 93 . . . . . . . . . . . . . . . . . . . . 21 (𝑓(𝑉 Walks 𝐸)𝑝 → ((𝑝‘0) = 𝐴 → ((𝑝‘(#‘𝑓)) = 𝐶 → ((#‘𝑓) = 2 → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))))
82813imp 1249 . . . . . . . . . . . . . . . . . . . 20 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → ((#‘𝑓) = 2 → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))
8382adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) → ((#‘𝑓) = 2 → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))
8483imp 444 . . . . . . . . . . . . . . . . . 18 ((((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) ∧ (#‘𝑓) = 2) → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
8539, 84sylbid 229 . . . . . . . . . . . . . . . . 17 ((((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) ∧ (#‘𝑓) = 2) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
8685ex 449 . . . . . . . . . . . . . . . 16 (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) → ((#‘𝑓) = 2 → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))
8786com23 84 . . . . . . . . . . . . . . 15 (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)) → ((#‘𝑓) = 2 → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))
8887ex 449 . . . . . . . . . . . . . 14 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)) → ((#‘𝑓) = 2 → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))))
8988com4t 91 . . . . . . . . . . . . 13 ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)) → ((#‘𝑓) = 2 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))))
9089ex 449 . . . . . . . . . . . 12 (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) → (((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶) → ((#‘𝑓) = 2 → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))))
9190com14 94 . . . . . . . . . . 11 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → (((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶) → ((#‘𝑓) = 2 → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) → (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))))
9291com23 84 . . . . . . . . . 10 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → ((#‘𝑓) = 2 → (((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) → (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))))
93923imp 1249 . . . . . . . . 9 (((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) → (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))
9493impcom 445 . . . . . . . 8 ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) → (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
9594com12 32 . . . . . . 7 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
9625adantr 480 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → 𝐴𝑉)
97 simpr 476 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → 𝑏𝑉)
9834adantr 480 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → 𝐶𝑉)
9996, 97, 983jca 1235 . . . . . . . . . . . . . . . 16 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → (𝐴𝑉𝑏𝑉𝐶𝑉))
100 otel3xp 5077 . . . . . . . . . . . . . . . 16 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝐴𝑉𝑏𝑉𝐶𝑉)) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉))
10199, 100sylan2 490 . . . . . . . . . . . . . . 15 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉)) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉))
102101ex 449 . . . . . . . . . . . . . 14 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))
103102adantr 480 . . . . . . . . . . . . 13 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))
104103com12 32 . . . . . . . . . . . 12 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))
105104adantll 746 . . . . . . . . . . 11 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))
106105imp 444 . . . . . . . . . 10 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉))
107 id 22 . . . . . . . . . . . . . . 15 (𝑓(𝑉 Walks 𝐸)𝑝𝑓(𝑉 Walks 𝐸)𝑝)
1081073ad2ant1 1075 . . . . . . . . . . . . . 14 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑓(𝑉 Walks 𝐸)𝑝)
109108adantl 481 . . . . . . . . . . . . 13 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑓(𝑉 Walks 𝐸)𝑝)
110109adantl 481 . . . . . . . . . . . 12 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → 𝑓(𝑉 Walks 𝐸)𝑝)
111 eqcom 2617 . . . . . . . . . . . . . . . . 17 (𝐴 = (𝑝‘0) ↔ (𝑝‘0) = 𝐴)
112111biimpi 205 . . . . . . . . . . . . . . . 16 (𝐴 = (𝑝‘0) → (𝑝‘0) = 𝐴)
1131123ad2ant1 1075 . . . . . . . . . . . . . . 15 ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘0) = 𝐴)
1141133ad2ant3 1077 . . . . . . . . . . . . . 14 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝‘0) = 𝐴)
115114adantl 481 . . . . . . . . . . . . 13 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑝‘0) = 𝐴)
116115adantl 481 . . . . . . . . . . . 12 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (𝑝‘0) = 𝐴)
117 eqcom 2617 . . . . . . . . . . . . . . . . . . 19 (𝐶 = (𝑝‘2) ↔ (𝑝‘2) = 𝐶)
118117biimpi 205 . . . . . . . . . . . . . . . . . 18 (𝐶 = (𝑝‘2) → (𝑝‘2) = 𝐶)
1191183ad2ant3 1077 . . . . . . . . . . . . . . . . 17 ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘2) = 𝐶)
120119, 61syl5ibr 235 . . . . . . . . . . . . . . . 16 ((#‘𝑓) = 2 → ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘(#‘𝑓)) = 𝐶))
121120a1i 11 . . . . . . . . . . . . . . 15 (𝑓(𝑉 Walks 𝐸)𝑝 → ((#‘𝑓) = 2 → ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘(#‘𝑓)) = 𝐶)))
1221213imp 1249 . . . . . . . . . . . . . 14 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝‘(#‘𝑓)) = 𝐶)
123122adantl 481 . . . . . . . . . . . . 13 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑝‘(#‘𝑓)) = 𝐶)
124123adantl 481 . . . . . . . . . . . 12 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (𝑝‘(#‘𝑓)) = 𝐶)
125110, 116, 1243jca 1235 . . . . . . . . . . 11 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶))
126 id 22 . . . . . . . . . . . . . 14 ((#‘𝑓) = 2 → (#‘𝑓) = 2)
1271263ad2ant2 1076 . . . . . . . . . . . . 13 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (#‘𝑓) = 2)
128127adantl 481 . . . . . . . . . . . 12 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (#‘𝑓) = 2)
129128adantl 481 . . . . . . . . . . 11 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (#‘𝑓) = 2)
13099adantll 746 . . . . . . . . . . . . . . . . 17 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → (𝐴𝑉𝑏𝑉𝐶𝑉))
131 oteqimp 7078 . . . . . . . . . . . . . . . . 17 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → ((𝐴𝑉𝑏𝑉𝐶𝑉) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = 𝑏 ∧ (2nd𝑇) = 𝐶)))
132130, 131syl5 33 . . . . . . . . . . . . . . . 16 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = 𝑏 ∧ (2nd𝑇) = 𝐶)))
133 eqeq2 2621 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑝‘1) → ((2nd ‘(1st𝑇)) = 𝑏 ↔ (2nd ‘(1st𝑇)) = (𝑝‘1)))
1341333anbi2d 1396 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑝‘1) → (((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = 𝑏 ∧ (2nd𝑇) = 𝐶) ↔ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)))
135134imbi2d 329 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑝‘1) → (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = 𝑏 ∧ (2nd𝑇) = 𝐶)) ↔ ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
136132, 135syl5ib 233 . . . . . . . . . . . . . . 15 (𝑏 = (𝑝‘1) → (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
1371363ad2ant2 1076 . . . . . . . . . . . . . 14 ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
1381373ad2ant3 1077 . . . . . . . . . . . . 13 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
139138impcom 445 . . . . . . . . . . . 12 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)))
140139impcom 445 . . . . . . . . . . 11 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))
141125, 129, 1403jca 1235 . . . . . . . . . 10 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)))
142106, 141jca 553 . . . . . . . . 9 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
143142ex 449 . . . . . . . 8 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)))))
144143rexlimdva 3013 . . . . . . 7 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)))))
14595, 144impbid 201 . . . . . 6 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
14624, 145bitrd 267 . . . . 5 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
1471462exbidv 1839 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (∃𝑓𝑝(𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) ↔ ∃𝑓𝑝𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
148 19.42vv 1907 . . . 4 (∃𝑓𝑝(𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) ↔ (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
149 rexcom4 3198 . . . . 5 (∃𝑏𝑉𝑓𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑓𝑏𝑉𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
150 rexcom4 3198 . . . . . 6 (∃𝑏𝑉𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑝𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
151150exbii 1764 . . . . 5 (∃𝑓𝑏𝑉𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑓𝑝𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
152149, 151bitr2i 264 . . . 4 (∃𝑓𝑝𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑏𝑉𝑓𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
153147, 148, 1523bitr3g 301 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) ↔ ∃𝑏𝑉𝑓𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
154 19.42vv 1907 . . . 4 (∃𝑓𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
155154rexbii 3023 . . 3 (∃𝑏𝑉𝑓𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
156153, 155syl6bb 275 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
15714, 156bitrd 267 1 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173  ⟨cotp 4133   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  1c1 9816  2c2 10947  #chash 12979   Walks cwalk 26026   WalkOn cwlkon 26030   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:  el2wlkonotot0  26399  el2wlksot  26407  frg2wot1  26584  frg2woteq  26587
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