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Theorem el2spthonot0 26398
 Description: A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 9-Mar-2018.)
Assertion
Ref Expression
el2spthonot0 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))))
Distinct variable groups:   𝐴,𝑏   𝐶,𝑏   𝐸,𝑏   𝑇,𝑏   𝑉,𝑏   𝑋,𝑏   𝑌,𝑏

Proof of Theorem el2spthonot0
Dummy variables 𝑓 𝑝 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 el2spthonot 26397 . 2 (((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))))))
2 fveq2 6103 . . . . . . . . . . . 12 (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → (1st𝑡) = (1st ‘⟨𝐴, 𝑏, 𝐶⟩))
32fveq2d 6107 . . . . . . . . . . 11 (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → (1st ‘(1st𝑡)) = (1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)))
43eqeq1d 2612 . . . . . . . . . 10 (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → ((1st ‘(1st𝑡)) = 𝐴 ↔ (1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴))
52fveq2d 6107 . . . . . . . . . . 11 (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → (2nd ‘(1st𝑡)) = (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)))
65eqeq1d 2612 . . . . . . . . . 10 (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → ((2nd ‘(1st𝑡)) = (𝑝‘1) ↔ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1)))
7 fveq2 6103 . . . . . . . . . . 11 (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → (2nd𝑡) = (2nd ‘⟨𝐴, 𝑏, 𝐶⟩))
87eqeq1d 2612 . . . . . . . . . 10 (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → ((2nd𝑡) = 𝐶 ↔ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))
94, 6, 83anbi123d 1391 . . . . . . . . 9 (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → (((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐶) ↔ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)))
1093anbi3d 1397 . . . . . . . 8 (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → ((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 ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
11102exbidv 1839 . . . . . . 7 (𝑡 = ⟨𝐴, 𝑏, 𝐶⟩ → (∃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 ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
1211elrab 3331 . . . . . 6 (⟨𝐴, 𝑏, 𝐶⟩ ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐶))} ↔ (⟨𝐴, 𝑏, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
1312a1i 11 . . . . 5 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → (⟨𝐴, 𝑏, 𝐶⟩ ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐶))} ↔ (⟨𝐴, 𝑏, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)))))
14 2spthonot 26393 . . . . . . 7 (((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))})
1514adantr 480 . . . . . 6 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → (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))})
1615eleq2d 2673 . . . . 5 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → (⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ⟨𝐴, 𝑏, 𝐶⟩ ∈ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐶))}))
17 simpr1 1060 . . . . . . . . . . . 12 ((((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → 𝑓(𝑉 SPaths 𝐸)𝑝)
18 id 22 . . . . . . . . . . . . . . . 16 ((𝑝‘0) = 𝐴 → (𝑝‘0) = 𝐴)
1918eqcoms 2618 . . . . . . . . . . . . . . 15 (𝐴 = (𝑝‘0) → (𝑝‘0) = 𝐴)
20193ad2ant1 1075 . . . . . . . . . . . . . 14 ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘0) = 𝐴)
21203ad2ant3 1077 . . . . . . . . . . . . 13 ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝‘0) = 𝐴)
2221adantl 481 . . . . . . . . . . . 12 ((((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑝‘0) = 𝐴)
23 fveq2 6103 . . . . . . . . . . . . . . 15 ((#‘𝑓) = 2 → (𝑝‘(#‘𝑓)) = (𝑝‘2))
24 id 22 . . . . . . . . . . . . . . . . 17 ((𝑝‘2) = 𝐶 → (𝑝‘2) = 𝐶)
2524eqcoms 2618 . . . . . . . . . . . . . . . 16 (𝐶 = (𝑝‘2) → (𝑝‘2) = 𝐶)
26253ad2ant3 1077 . . . . . . . . . . . . . . 15 ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → (𝑝‘2) = 𝐶)
2723, 26sylan9eq 2664 . . . . . . . . . . . . . 14 (((#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝‘(#‘𝑓)) = 𝐶)
28273adant1 1072 . . . . . . . . . . . . 13 ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → (𝑝‘(#‘𝑓)) = 𝐶)
2928adantl 481 . . . . . . . . . . . 12 ((((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑝‘(#‘𝑓)) = 𝐶)
3017, 22, 293jca 1235 . . . . . . . . . . 11 ((((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶))
31 simpr2 1061 . . . . . . . . . . 11 ((((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (#‘𝑓) = 2)
32 eqidd 2611 . . . . . . . . . . . . . 14 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → ⟨𝐴, 𝑏, 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩)
33 simpl 472 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐶𝑉) → 𝐴𝑉)
3433adantr 480 . . . . . . . . . . . . . 14 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → 𝐴𝑉)
35 simpr 476 . . . . . . . . . . . . . 14 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → 𝑏𝑉)
36 simpr 476 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐶𝑉) → 𝐶𝑉)
3736adantr 480 . . . . . . . . . . . . . 14 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → 𝐶𝑉)
38 oteqimp 7078 . . . . . . . . . . . . . . 15 (⟨𝐴, 𝑏, 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩ → ((𝐴𝑉𝑏𝑉𝐶𝑉) → ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)))
3938imp 444 . . . . . . . . . . . . . 14 ((⟨𝐴, 𝑏, 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝐴𝑉𝑏𝑉𝐶𝑉)) → ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))
4032, 34, 35, 37, 39syl13anc 1320 . . . . . . . . . . . . 13 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))
4140adantr 480 . . . . . . . . . . . 12 ((((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))
42 eqeq2 2621 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑝‘1) → ((2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ↔ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1)))
43423ad2ant2 1076 . . . . . . . . . . . . . . 15 ((𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)) → ((2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ↔ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1)))
44433ad2ant3 1077 . . . . . . . . . . . . . 14 ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) → ((2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ↔ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1)))
4544adantl 481 . . . . . . . . . . . . 13 ((((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → ((2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ↔ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1)))
46453anbi2d 1396 . . . . . . . . . . . 12 ((((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → (((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏 ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶) ↔ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)))
4741, 46mpbid 221 . . . . . . . . . . 11 ((((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))
4830, 31, 473jca 1235 . . . . . . . . . 10 ((((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) ∧ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)))
49 simpr11 1138 . . . . . . . . . . 11 ((((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) ∧ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))) → 𝑓(𝑉 SPaths 𝐸)𝑝)
50 simpr2 1061 . . . . . . . . . . 11 ((((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) ∧ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))) → (#‘𝑓) = 2)
5123eqeq1d 2612 . . . . . . . . . . . . . . . 16 ((#‘𝑓) = 2 → ((𝑝‘(#‘𝑓)) = 𝐶 ↔ (𝑝‘2) = 𝐶))
52513anbi3d 1397 . . . . . . . . . . . . . . 15 ((#‘𝑓) = 2 → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ↔ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶)))
53 fvex 6113 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝‘0) ∈ V
54 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑝‘0) = 𝐴 → ((𝑝‘0) ∈ V ↔ 𝐴 ∈ V))
5553, 54mpbii 222 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝‘0) = 𝐴𝐴 ∈ V)
5655adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → 𝐴 ∈ V)
57 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑏 ∈ V
5857a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → 𝑏 ∈ V)
59 fvex 6113 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝‘2) ∈ V
60 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑝‘2) = 𝐶 → ((𝑝‘2) ∈ V ↔ 𝐶 ∈ V))
6159, 60mpbii 222 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝‘2) = 𝐶𝐶 ∈ V)
6261adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → 𝐶 ∈ V)
63 ot2ndg 7074 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ V ∧ 𝑏 ∈ V ∧ 𝐶 ∈ V) → (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏)
6456, 58, 62, 63syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝑏)
6564eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . 21 (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → ((2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ↔ 𝑏 = (𝑝‘1)))
66 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐴 = (𝑝‘0) → 𝐴 = (𝑝‘0))
6766eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑝‘0) = 𝐴𝐴 = (𝑝‘0))
6867adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → 𝐴 = (𝑝‘0))
6968adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) ∧ 𝑏 = (𝑝‘1)) → 𝐴 = (𝑝‘0))
70 simpr 476 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) ∧ 𝑏 = (𝑝‘1)) → 𝑏 = (𝑝‘1))
71 id 22 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐶 = (𝑝‘2) → 𝐶 = (𝑝‘2))
7271eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑝‘2) = 𝐶𝐶 = (𝑝‘2))
7372ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) ∧ 𝑏 = (𝑝‘1)) → 𝐶 = (𝑝‘2))
7469, 70, 733jca 1235 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) ∧ 𝑏 = (𝑝‘1)) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))
7574ex 449 . . . . . . . . . . . . . . . . . . . . 21 (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (𝑏 = (𝑝‘1) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
7665, 75sylbid 229 . . . . . . . . . . . . . . . . . . . 20 (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → ((2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
7776com12 32 . . . . . . . . . . . . . . . . . . 19 ((2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) → (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
78773ad2ant2 1076 . . . . . . . . . . . . . . . . . 18 (((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶) → (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
7978com12 32 . . . . . . . . . . . . . . . . 17 (((𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
80793adant1 1072 . . . . . . . . . . . . . . . 16 ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
8180a1i 11 . . . . . . . . . . . . . . 15 ((#‘𝑓) = 2 → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘2) = 𝐶) → (((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
8252, 81sylbid 229 . . . . . . . . . . . . . 14 ((#‘𝑓) = 2 → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → (((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
8382com12 32 . . . . . . . . . . . . 13 ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) → ((#‘𝑓) = 2 → (((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
84833imp 1249 . . . . . . . . . . . 12 (((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))
8584adantl 481 . . . . . . . . . . 11 ((((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) ∧ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))) → (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))
8649, 50, 853jca 1235 . . . . . . . . . 10 ((((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) ∧ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))) → (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
8748, 86impbida 873 . . . . . . . . 9 (((𝐴𝑉𝐶𝑉) ∧ 𝑏𝑉) → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
8887adantll 746 . . . . . . . 8 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
89 vex 3176 . . . . . . . . . . . . 13 𝑓 ∈ V
90 vex 3176 . . . . . . . . . . . . 13 𝑝 ∈ V
9189, 90pm3.2i 470 . . . . . . . . . . . 12 (𝑓 ∈ V ∧ 𝑝 ∈ V)
92 isspthonpth 26114 . . . . . . . . . . . 12 (((𝑉𝑋𝐸𝑌) ∧ (𝑓 ∈ V ∧ 𝑝 ∈ V) ∧ (𝐴𝑉𝐶𝑉)) → (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶)))
9391, 92mp3an2 1404 . . . . . . . . . . 11 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶)))
9493adantr 480 . . . . . . . . . 10 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ↔ (𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶)))
9594bicomd 212 . . . . . . . . 9 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ↔ 𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝))
96953anbi1d 1395 . . . . . . . 8 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → (((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐶) ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)) ↔ (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
9788, 96bitrd 267 . . . . . . 7 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ (𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
98972exbidv 1839 . . . . . 6 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → (∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶))))
99 eqidd 2611 . . . . . . . 8 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ⟨𝐴, 𝑏, 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩)
10033ad2antlr 759 . . . . . . . 8 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → 𝐴𝑉)
101 simpr 476 . . . . . . . 8 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → 𝑏𝑉)
10236ad2antlr 759 . . . . . . . 8 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → 𝐶𝑉)
103 otel3xp 5077 . . . . . . . 8 ((⟨𝐴, 𝑏, 𝐶⟩ = ⟨𝐴, 𝑏, 𝐶⟩ ∧ (𝐴𝑉𝑏𝑉𝐶𝑉)) → ⟨𝐴, 𝑏, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉))
10499, 100, 101, 102, 103syl13anc 1320 . . . . . . 7 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ⟨𝐴, 𝑏, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉))
105104biantrurd 528 . . . . . 6 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → (∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)) ↔ (⟨𝐴, 𝑏, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)))))
10698, 105bitrd 267 . . . . 5 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → (∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ (⟨𝐴, 𝑏, 𝐶⟩ ∈ ((𝑉 × 𝑉) × 𝑉) ∧ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐶)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = 𝐴 ∧ (2nd ‘(1st ‘⟨𝐴, 𝑏, 𝐶⟩)) = (𝑝‘1) ∧ (2nd ‘⟨𝐴, 𝑏, 𝐶⟩) = 𝐶)))))
10713, 16, 1063bitr4rd 300 . . . 4 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → (∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))) ↔ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶)))
108107anbi2d 736 . . 3 ((((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) ∧ 𝑏𝑉) → ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))))
109108rexbidva 3031 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))))
1101, 109bitrd 267 1 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))))
 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   SPaths cspath 26029   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:  usg2spthonot1  26417
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