MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgr2wwlk1 Structured version   Visualization version   GIF version

Theorem frgr2wwlk1 28041
Description: In a friendship graph, there is exactly one walk of length 2 between two different vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.) (Revised by AV, 13-May-2021.) (Proof shortened by AV, 16-Mar-2022.)
Hypothesis
Ref Expression
frgr2wwlkeu.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
frgr2wwlk1 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (♯‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1)

Proof of Theorem frgr2wwlk1
Dummy variables 𝑐 𝑑 𝑡 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgr2wwlkeu.v . . . 4 𝑉 = (Vtx‘𝐺)
21frgr2wwlkn0 28040 . . 3 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅)
31elwwlks2ons3 27667 . . . . . 6 (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
41elwwlks2ons3 27667 . . . . . 6 (𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
53, 4anbi12i 626 . . . . 5 ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ↔ (∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ∧ ∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
61frgr2wwlkeu 28039 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑥𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
7 s3eq2 14227 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑦𝐵”⟩)
87eleq1d 2902 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
98reu4 3726 . . . . . . . . . . . 12 (∃!𝑥𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ (∃𝑥𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ∀𝑥𝑉𝑦𝑉 ((⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑥 = 𝑦)))
10 s3eq2 14227 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑑 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
1110eleq1d 2902 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑑 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
1211anbi1d 629 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑑 → ((⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ↔ (⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
13 equequ1 2025 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑑 → (𝑥 = 𝑦𝑑 = 𝑦))
1412, 13imbi12d 346 . . . . . . . . . . . . . . 15 (𝑥 = 𝑑 → (((⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑥 = 𝑦) ↔ ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑦)))
15 s3eq2 14227 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑐 → ⟨“𝐴𝑦𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩)
1615eleq1d 2902 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑐 → (⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
1716anbi2d 628 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑐 → ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ↔ (⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
18 equequ2 2026 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑐 → (𝑑 = 𝑦𝑑 = 𝑐))
1917, 18imbi12d 346 . . . . . . . . . . . . . . 15 (𝑦 = 𝑐 → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑦) ↔ ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐)))
2014, 19rspc2va 3638 . . . . . . . . . . . . . 14 (((𝑑𝑉𝑐𝑉) ∧ ∀𝑥𝑉𝑦𝑉 ((⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑥 = 𝑦)) → ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐))
21 pm3.35 799 . . . . . . . . . . . . . . . . . . . . . . 23 (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ∧ ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐)) → 𝑑 = 𝑐)
22 s3eq2 14227 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 = 𝑑 → ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
2322equcoms 2020 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑑 = 𝑐 → ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
2423adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑 = 𝑐 ∧ (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩)) → ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
25 eqeq12 2840 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩) → (𝑡 = 𝑤 ↔ ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩))
2625adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑 = 𝑐 ∧ (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩)) → (𝑡 = 𝑤 ↔ ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩))
2724, 26mpbird 258 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑑 = 𝑐 ∧ (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩)) → 𝑡 = 𝑤)
2827equcomd 2019 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑑 = 𝑐 ∧ (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩)) → 𝑤 = 𝑡)
2928ex 413 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 = 𝑐 → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩) → 𝑤 = 𝑡))
3021, 29syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ∧ ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐)) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩) → 𝑤 = 𝑡))
3130ex 413 . . . . . . . . . . . . . . . . . . . . 21 ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩) → 𝑤 = 𝑡)))
3231com23 86 . . . . . . . . . . . . . . . . . . . 20 ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩) → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → 𝑤 = 𝑡)))
3332exp4b 431 . . . . . . . . . . . . . . . . . . 19 (⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (𝑡 = ⟨“𝐴𝑐𝐵”⟩ → (𝑤 = ⟨“𝐴𝑑𝐵”⟩ → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → 𝑤 = 𝑡)))))
3433com13 88 . . . . . . . . . . . . . . . . . 18 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ → (⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (𝑤 = ⟨“𝐴𝑑𝐵”⟩ → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → 𝑤 = 𝑡)))))
3534imp 407 . . . . . . . . . . . . . . . . 17 ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (𝑤 = ⟨“𝐴𝑑𝐵”⟩ → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → 𝑤 = 𝑡))))
3635com13 88 . . . . . . . . . . . . . . . 16 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ → (⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → 𝑤 = 𝑡))))
3736imp 407 . . . . . . . . . . . . . . 15 ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → 𝑤 = 𝑡)))
3837com13 88 . . . . . . . . . . . . . 14 (((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
3920, 38syl 17 . . . . . . . . . . . . 13 (((𝑑𝑉𝑐𝑉) ∧ ∀𝑥𝑉𝑦𝑉 ((⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑥 = 𝑦)) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
4039expcom 414 . . . . . . . . . . . 12 (∀𝑥𝑉𝑦𝑉 ((⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑥 = 𝑦) → ((𝑑𝑉𝑐𝑉) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))))
419, 40simplbiim 505 . . . . . . . . . . 11 (∃!𝑥𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → ((𝑑𝑉𝑐𝑉) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))))
4241impl 456 . . . . . . . . . 10 (((∃!𝑥𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑑𝑉) ∧ 𝑐𝑉) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
4342rexlimdva 3289 . . . . . . . . 9 ((∃!𝑥𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑑𝑉) → (∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
4443com23 86 . . . . . . . 8 ((∃!𝑥𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑑𝑉) → ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
4544rexlimdva 3289 . . . . . . 7 (∃!𝑥𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
4645impd 411 . . . . . 6 (∃!𝑥𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → ((∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ∧ ∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))) → 𝑤 = 𝑡))
476, 46syl 17 . . . . 5 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ∧ ∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))) → 𝑤 = 𝑡))
485, 47syl5bi 243 . . . 4 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))
4948alrimivv 1922 . . 3 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∀𝑤𝑡((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))
50 eqeuel 4326 . . 3 (((𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅ ∧ ∀𝑤𝑡((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)) → ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
512, 49, 50syl2anc 584 . 2 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
52 ovex 7183 . . 3 (𝐴(2 WWalksNOn 𝐺)𝐵) ∈ V
53 euhash1 13776 . . 3 ((𝐴(2 WWalksNOn 𝐺)𝐵) ∈ V → ((♯‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1 ↔ ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
5452, 53mp1i 13 . 2 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((♯‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1 ↔ ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
5551, 54mpbird 258 1 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (♯‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081  wal 1528   = wceq 1530  wcel 2107  ∃!weu 2651  wne 3021  wral 3143  wrex 3144  ∃!wreu 3145  Vcvv 3500  c0 4295  cfv 6354  (class class class)co 7150  1c1 10532  2c2 11686  chash 13685  ⟨“cs3 14199  Vtxcvtx 26714   WWalksNOn cwwlksnon 27538   FriendGraph cfrgr 27970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-ac2 9879  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ifp 1057  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8284  df-map 8403  df-pm 8404  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-dju 9324  df-card 9362  df-ac 9536  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-xnn0 11962  df-z 11976  df-uz 12238  df-fz 12888  df-fzo 13029  df-hash 13686  df-word 13857  df-concat 13918  df-s1 13945  df-s2 14205  df-s3 14206  df-edg 26766  df-uhgr 26776  df-upgr 26800  df-umgr 26801  df-usgr 26869  df-wlks 27314  df-wwlks 27541  df-wwlksn 27542  df-wwlksnon 27543  df-frgr 27971
This theorem is referenced by:  frgr2wsp1  28042
  Copyright terms: Public domain W3C validator