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Theorem el2spthonot 26397
 Description: A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
Assertion
Ref Expression
el2spthonot (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 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 el2spthonot
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 2spthonot 26393 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐶))})
21eleq2d 2673 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ 𝑇 ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 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 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐶)) ↔ (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
12112exbidv 1839 . . . 4 (𝑡 = 𝑇 → (∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐶)) ↔ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 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 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐶))} ↔ (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
142, 13syl6bb 275 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)))))
15 vex 3176 . . . . . . . . . 10 𝑓 ∈ V
16 vex 3176 . . . . . . . . . 10 𝑝 ∈ V
1715, 16pm3.2i 470 . . . . . . . . 9 (𝑓 ∈ V ∧ 𝑝 ∈ V)
18 isspthon 26113 . . . . . . . . 9 (((𝑉𝑋𝐸𝑌) ∧ (𝑓 ∈ V ∧ 𝑝 ∈ V) ∧ (𝐴𝑉𝐶𝑉)) → (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝)))
1917, 18mp3an2 1404 . . . . . . . 8 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝)))
20193anbi1d 1395 . . . . . . 7 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)) ↔ ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
2120anbi2d 736 . . . . . 6 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) ↔ (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)))))
22 simpl 472 . . . . . . . . . . . . . . . 16 ((𝐴𝑉𝐶𝑉) → 𝐴𝑉)
2322ad2antll 761 . . . . . . . . . . . . . . 15 ((((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) → 𝐴𝑉)
24 spthispth 26103 . . . . . . . . . . . . . . . . . . 19 (𝑓(𝑉 SPaths 𝐸)𝑝𝑓(𝑉 Paths 𝐸)𝑝)
25 pthistrl 26102 . . . . . . . . . . . . . . . . . . 19 (𝑓(𝑉 Paths 𝐸)𝑝𝑓(𝑉 Trails 𝐸)𝑝)
26 trliswlk 26069 . . . . . . . . . . . . . . . . . . 19 (𝑓(𝑉 Trails 𝐸)𝑝𝑓(𝑉 Walks 𝐸)𝑝)
27 el2wlkonotlem 26389 . . . . . . . . . . . . . . . . . . . 20 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → (𝑝‘1) ∈ 𝑉)
2827ex 449 . . . . . . . . . . . . . . . . . . 19 (𝑓(𝑉 Walks 𝐸)𝑝 → ((#‘𝑓) = 2 → (𝑝‘1) ∈ 𝑉))
2924, 25, 26, 284syl 19 . . . . . . . . . . . . . . . . . 18 (𝑓(𝑉 SPaths 𝐸)𝑝 → ((#‘𝑓) = 2 → (𝑝‘1) ∈ 𝑉))
3029adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) → ((#‘𝑓) = 2 → (𝑝‘1) ∈ 𝑉))
3130imp 444 . . . . . . . . . . . . . . . 16 (((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) → (𝑝‘1) ∈ 𝑉)
3231adantr 480 . . . . . . . . . . . . . . 15 ((((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) → (𝑝‘1) ∈ 𝑉)
33 simpr 476 . . . . . . . . . . . . . . . 16 ((𝐴𝑉𝐶𝑉) → 𝐶𝑉)
3433ad2antll 761 . . . . . . . . . . . . . . 15 ((((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) → 𝐶𝑉)
35 el2xptp0 7103 . . . . . . . . . . . . . . 15 ((𝐴𝑉 ∧ (𝑝‘1) ∈ 𝑉𝐶𝑉) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)) ↔ 𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩))
3623, 32, 34, 35syl3anc 1318 . . . . . . . . . . . . . 14 ((((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)) ↔ 𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩))
37 oteq2 4350 . . . . . . . . . . . . . . . . . . . 20 ((𝑝‘1) = 𝑏 → ⟨𝐴, (𝑝‘1), 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩)
3837eqcoms 2618 . . . . . . . . . . . . . . . . . . 19 (𝑏 = (𝑝‘1) → ⟨𝐴, (𝑝‘1), 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩)
3938eqeq2d 2620 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑝‘1) → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ ↔ 𝑇 = ⟨𝐴, 𝑏, 𝐶⟩))
4039biimpd 218 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑝‘1) → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → 𝑇 = ⟨𝐴, 𝑏, 𝐶⟩))
4140adantl 481 . . . . . . . . . . . . . . . 16 (((((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) ∧ 𝑏 = (𝑝‘1)) → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → 𝑇 = ⟨𝐴, 𝑏, 𝐶⟩))
42 simpllr 795 . . . . . . . . . . . . . . . . . 18 ((((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) → 𝑓(𝑉 SPaths 𝐸)𝑝)
4342adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) ∧ 𝑏 = (𝑝‘1)) → 𝑓(𝑉 SPaths 𝐸)𝑝)
44 simpllr 795 . . . . . . . . . . . . . . . . 17 (((((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) ∧ 𝑏 = (𝑝‘1)) → (#‘𝑓) = 2)
45 iswlkon 26062 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑉𝑋𝐸𝑌) ∧ (𝑓 ∈ V ∧ 𝑝 ∈ V) ∧ (𝐴𝑉𝐶𝑉)) → (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶)))
4617, 45mp3an2 1404 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶)))
47 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((#‘𝑓) = 2 → (𝑝‘(#‘𝑓)) = (𝑝‘2))
4847eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((#‘𝑓) = 2 → ((𝑝‘(#‘𝑓)) = 𝐶 ↔ (𝑝‘2) = 𝐶))
4948anbi2d 736 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((#‘𝑓) = 2 → (((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ↔ ((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶)))
50 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝‘0) = 𝐴𝐴 = (𝑝‘0))
5150biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑝‘0) = 𝐴𝐴 = (𝑝‘0))
5251adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → 𝐴 = (𝑝‘0))
5352adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) ∧ 𝑏 = (𝑝‘1)) → 𝐴 = (𝑝‘0))
54 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) ∧ 𝑏 = (𝑝‘1)) → 𝑏 = (𝑝‘1))
55 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑝‘2) = 𝐶𝐶 = (𝑝‘2))
5655biimpi 205 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑝‘2) = 𝐶𝐶 = (𝑝‘2))
5756adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → 𝐶 = (𝑝‘2))
5857adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) ∧ 𝑏 = (𝑝‘1)) → 𝐶 = (𝑝‘2))
5953, 54, 583jca 1235 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) ∧ 𝑏 = (𝑝‘1)) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))
6059ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (𝑏 = (𝑝‘1) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
6160a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((#‘𝑓) = 2 → (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (𝑏 = (𝑝‘1) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
6249, 61sylbid 229 . . . . . . . . . . . . . . . . . . . . . . . 24 ((#‘𝑓) = 2 → (((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → (𝑏 = (𝑝‘1) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
6362com12 32 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → ((#‘𝑓) = 2 → (𝑏 = (𝑝‘1) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
64633adant1 1072 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → ((#‘𝑓) = 2 → (𝑏 = (𝑝‘1) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
6564a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → ((#‘𝑓) = 2 → (𝑏 = (𝑝‘1) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
6646, 65sylbid 229 . . . . . . . . . . . . . . . . . . . 20 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 → ((#‘𝑓) = 2 → (𝑏 = (𝑝‘1) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
6766com3l 87 . . . . . . . . . . . . . . . . . . 19 (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 → ((#‘𝑓) = 2 → (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑏 = (𝑝‘1) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
6867adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) → ((#‘𝑓) = 2 → (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑏 = (𝑝‘1) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
6968imp41 617 . . . . . . . . . . . . . . . . 17 (((((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) ∧ 𝑏 = (𝑝‘1)) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))
7043, 44, 693jca 1235 . . . . . . . . . . . . . . . 16 (((((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) ∧ 𝑏 = (𝑝‘1)) → (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
7141, 70jctird 565 . . . . . . . . . . . . . . 15 (((((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) ∧ 𝑏 = (𝑝‘1)) → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
7232, 71rspcimedv 3284 . . . . . . . . . . . . . 14 ((((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) → (𝑇 = ⟨𝐴, (𝑝‘1), 𝐶⟩ → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
7336, 72sylbid 229 . . . . . . . . . . . . 13 ((((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) ∧ ((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉))) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
7473exp4b 630 . . . . . . . . . . . 12 (((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) → (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) → (((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))))
7574com24 93 . . . . . . . . . . 11 (((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2) → (((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) → (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))))
7675ex 449 . . . . . . . . . 10 ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) → ((#‘𝑓) = 2 → (((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) → (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))))
77763imp 1249 . . . . . . . . 9 (((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) → (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))))
7877impcom 445 . . . . . . . 8 ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) → (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
7978com12 32 . . . . . . 7 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) → ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
8022adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → 𝐴𝑉)
81 simpr 476 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → 𝑏𝑉)
8233adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → 𝐶𝑉)
8380, 81, 823jca 1235 . . . . . . . . . . . . . . . . 17 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → (𝐴𝑉𝑏𝑉𝐶𝑉))
84 otel3xp 5077 . . . . . . . . . . . . . . . . 17 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝐴𝑉𝑏𝑉𝐶𝑉)) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉))
8583, 84sylan2 490 . . . . . . . . . . . . . . . 16 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉)) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉))
8685ex 449 . . . . . . . . . . . . . . 15 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))
8786adantr 480 . . . . . . . . . . . . . 14 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))
8887com12 32 . . . . . . . . . . . . 13 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))
8988ex 449 . . . . . . . . . . . 12 ((𝐴𝑉𝐶𝑉) → (𝑏𝑉 → ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉))))
9089adantl 481 . . . . . . . . . . 11 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑏𝑉 → ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉))))
9190imp31 447 . . . . . . . . . 10 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉))
9224, 25, 263syl 18 . . . . . . . . . . . . . . 15 (𝑓(𝑉 SPaths 𝐸)𝑝𝑓(𝑉 Walks 𝐸)𝑝)
93923ad2ant1 1075 . . . . . . . . . . . . . 14 ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑓(𝑉 Walks 𝐸)𝑝)
9493ad2antll 761 . . . . . . . . . . . . 13 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → 𝑓(𝑉 Walks 𝐸)𝑝)
95 eqcom 2617 . . . . . . . . . . . . . . . . 17 (𝐴 = (𝑝‘0) ↔ (𝑝‘0) = 𝐴)
9695biimpi 205 . . . . . . . . . . . . . . . 16 (𝐴 = (𝑝‘0) → (𝑝‘0) = 𝐴)
97963ad2ant1 1075 . . . . . . . . . . . . . . 15 ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘0) = 𝐴)
98973ad2ant3 1077 . . . . . . . . . . . . . 14 ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝‘0) = 𝐴)
9998ad2antll 761 . . . . . . . . . . . . 13 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (𝑝‘0) = 𝐴)
100 eqcom 2617 . . . . . . . . . . . . . . . . . . 19 (𝐶 = (𝑝‘2) ↔ (𝑝‘2) = 𝐶)
101100biimpi 205 . . . . . . . . . . . . . . . . . 18 (𝐶 = (𝑝‘2) → (𝑝‘2) = 𝐶)
1021013ad2ant3 1077 . . . . . . . . . . . . . . . . 17 ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘2) = 𝐶)
103102, 48syl5ibr 235 . . . . . . . . . . . . . . . 16 ((#‘𝑓) = 2 → ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘(#‘𝑓)) = 𝐶))
104103a1i 11 . . . . . . . . . . . . . . 15 (𝑓(𝑉 SPaths 𝐸)𝑝 → ((#‘𝑓) = 2 → ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘(#‘𝑓)) = 𝐶)))
1051043imp 1249 . . . . . . . . . . . . . 14 ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝‘(#‘𝑓)) = 𝐶)
106105ad2antll 761 . . . . . . . . . . . . 13 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (𝑝‘(#‘𝑓)) = 𝐶)
10746adantr 480 . . . . . . . . . . . . . 14 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶)))
108107adantr 480 . . . . . . . . . . . . 13 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶)))
10994, 99, 106, 108mpbir3and 1238 . . . . . . . . . . . 12 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → 𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝)
110 id 22 . . . . . . . . . . . . . 14 (𝑓(𝑉 SPaths 𝐸)𝑝𝑓(𝑉 SPaths 𝐸)𝑝)
1111103ad2ant1 1075 . . . . . . . . . . . . 13 ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → 𝑓(𝑉 SPaths 𝐸)𝑝)
112111ad2antll 761 . . . . . . . . . . . 12 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → 𝑓(𝑉 SPaths 𝐸)𝑝)
113109, 112jca 553 . . . . . . . . . . 11 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝))
114 id 22 . . . . . . . . . . . . 13 ((#‘𝑓) = 2 → (#‘𝑓) = 2)
1151143ad2ant2 1076 . . . . . . . . . . . 12 ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (#‘𝑓) = 2)
116115ad2antll 761 . . . . . . . . . . 11 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (#‘𝑓) = 2)
11722adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → 𝐴𝑉)
118117adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → 𝐴𝑉)
119 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → 𝑏𝑉)
12033adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → 𝐶𝑉)
121120adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → 𝐶𝑉)
122118, 119, 1213jca 1235 . . . . . . . . . . . . . . . . . 18 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → (𝐴𝑉𝑏𝑉𝐶𝑉))
123 oteqimp 7078 . . . . . . . . . . . . . . . . . . 19 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → ((𝐴𝑉𝑏𝑉𝐶𝑉) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = 𝑏 ∧ (2nd𝑇) = 𝐶)))
124123imp 444 . . . . . . . . . . . . . . . . . 18 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝐴𝑉𝑏𝑉𝐶𝑉)) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = 𝑏 ∧ (2nd𝑇) = 𝐶))
125122, 124sylan2 490 . . . . . . . . . . . . . . . . 17 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉)) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = 𝑏 ∧ (2nd𝑇) = 𝐶))
126125ex 449 . . . . . . . . . . . . . . . 16 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = 𝑏 ∧ (2nd𝑇) = 𝐶)))
127 eqeq2 2621 . . . . . . . . . . . . . . . . . . 19 ((𝑝‘1) = 𝑏 → ((2nd ‘(1st𝑇)) = (𝑝‘1) ↔ (2nd ‘(1st𝑇)) = 𝑏))
128127eqcoms 2618 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑝‘1) → ((2nd ‘(1st𝑇)) = (𝑝‘1) ↔ (2nd ‘(1st𝑇)) = 𝑏))
1291283anbi2d 1396 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑝‘1) → (((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶) ↔ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = 𝑏 ∧ (2nd𝑇) = 𝐶)))
130129imbi2d 329 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑝‘1) → (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)) ↔ ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = 𝑏 ∧ (2nd𝑇) = 𝐶))))
131126, 130syl5ibr 235 . . . . . . . . . . . . . . 15 (𝑏 = (𝑝‘1) → (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ → ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
1321313ad2ant2 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))))
1331323ad2ant3 1077 . . . . . . . . . . . . 13 ((𝑓(𝑉 SPaths 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))))
134133impcom 445 . . . . . . . . . . . 12 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 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)))
135134impcom 445 . . . . . . . . . . 11 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))
136113, 116, 1353jca 1235 . . . . . . . . . 10 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)))
13791, 136jca 553 . . . . . . . . 9 (((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) ∧ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
138137ex 449 . . . . . . . 8 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)))))
139138rexlimdva 3013 . . . . . . 7 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶)))))
14079, 139impbid 201 . . . . . 6 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐶)𝑝𝑓(𝑉 SPaths 𝐸)𝑝) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
14121, 140bitrd 267 . . . . 5 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
1421412exbidv 1839 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (∃𝑓𝑝(𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) ↔ ∃𝑓𝑝𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
143 19.42vv 1907 . . . 4 (∃𝑓𝑝(𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) ↔ (𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))))
144 rexcom4 3198 . . . . 5 (∃𝑏𝑉𝑓𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑓𝑏𝑉𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
145 rexcom4 3198 . . . . . 6 (∃𝑏𝑉𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑝𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
146145exbii 1764 . . . . 5 (∃𝑓𝑏𝑉𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑓𝑝𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
147144, 146bitr2i 264 . . . 4 (∃𝑓𝑝𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑏𝑉𝑓𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
148142, 143, 1473bitr3g 301 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) ↔ ∃𝑏𝑉𝑓𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
149 19.42vv 1907 . . . 4 (∃𝑓𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
150149rexbii 3023 . . 3 (∃𝑏𝑉𝑓𝑝(𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
151148, 150syl6bb 275 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → ((𝑇 ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑇)) = 𝐴 ∧ (2nd ‘(1st𝑇)) = (𝑝‘1) ∧ (2nd𝑇) = 𝐶))) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))
15214, 151bitrd 267 1 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 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   Trails ctrail 26027   Paths cpath 26028   SPaths cspath 26029   WalkOn cwlkon 26030   SPathOn cspthon 26033   2SPathOnOt c2pthonot 26384 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-wlk 26036  df-trail 26037  df-pth 26038  df-spth 26039  df-wlkon 26042  df-spthon 26045  df-2spthonot 26387 This theorem is referenced by:  el2spthonot0  26398  el2pthsot  26408  2spontn0vne  26414  2spotiundisj  26589
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