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Theorem frgr2wwlk1 27500
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 27499 . . 3 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅)
31elwwlks2ons3 27091 . . . . . 6 (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
41elwwlks2ons3 27091 . . . . . 6 (𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
53, 4anbi12i 614 . . . . 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 27498 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑥𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
7 s3eq2 13835 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑦𝐵”⟩)
87eleq1d 2869 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
98reu4 3595 . . . . . . . . . . . 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 13835 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑑 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
1110eleq1d 2869 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑑 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
1211anbi1d 617 . . . . . . . . . . . . . . . 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 2123 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑑 → (𝑥 = 𝑦𝑑 = 𝑦))
1412, 13imbi12d 335 . . . . . . . . . . . . . . 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 13835 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑐 → ⟨“𝐴𝑦𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩)
1615eleq1d 2869 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑐 → (⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
1716anbi2d 616 . . . . . . . . . . . . . . . 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 2124 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑐 → (𝑑 = 𝑦𝑑 = 𝑐))
1917, 18imbi12d 335 . . . . . . . . . . . . . . 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 3515 . . . . . . . . . . . . . 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 pm3.35 828 . . . . . . . . . . . . . . . . . . . . . . 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 13835 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 = 𝑑 → ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
2322equcoms 2118 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑑 = 𝑐 → ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
2423adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑 = 𝑐 ∧ (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩)) → ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
25 eqeq12 2818 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩) → (𝑡 = 𝑤 ↔ ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩))
2625adantl 469 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑑 = 𝑐 ∧ (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩)) → (𝑡 = 𝑤 ↔ ⟨“𝐴𝑐𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩))
2724, 26mpbird 248 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑑 = 𝑐 ∧ (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩)) → 𝑡 = 𝑤)
2827equcomd 2117 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑑 = 𝑐 ∧ (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ 𝑤 = ⟨“𝐴𝑑𝐵”⟩)) → 𝑤 = 𝑡)
2928ex 399 . . . . . . . . . . . . . . . . . . . . . . 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 399 . . . . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . . . . 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 400 . . . . . . . . . . . 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 495 . . . . . . . . . . 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 445 . . . . . . . . . 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 3218 . . . . . . . . 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 3218 . . . . . . 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 398 . . . . . 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 233 . . . 4 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))
4948alrimivv 2021 . . 3 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∀𝑤𝑡((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))
50 eqeuel 4139 . . 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 575 . 2 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
52 ovex 6903 . . 3 (𝐴(2 WWalksNOn 𝐺)𝐵) ∈ V
53 euhash1 13421 . . 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 248 1 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (♯‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100  wal 1635   = wceq 1637  wcel 2158  ∃!weu 2632  wne 2977  wral 3095  wrex 3096  ∃!wreu 3097  Vcvv 3390  c0 4113  cfv 6098  (class class class)co 6871  1c1 10219  2c2 11352  chash 13333  ⟨“cs3 13807  Vtxcvtx 26084   WWalksNOn cwwlksnon 26944   FriendGraph cfrgr 27427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-rep 4960  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176  ax-ac2 9567  ax-cnex 10274  ax-resscn 10275  ax-1cn 10276  ax-icn 10277  ax-addcl 10278  ax-addrcl 10279  ax-mulcl 10280  ax-mulrcl 10281  ax-mulcom 10282  ax-addass 10283  ax-mulass 10284  ax-distr 10285  ax-i2m1 10286  ax-1ne0 10287  ax-1rid 10288  ax-rnegex 10289  ax-rrecex 10290  ax-cnre 10291  ax-pre-lttri 10292  ax-pre-lttrn 10293  ax-pre-ltadd 10294  ax-pre-mulgt0 10295
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-ifp 1079  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-nel 3081  df-ral 3100  df-rex 3101  df-reu 3102  df-rmo 3103  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4627  df-int 4666  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-tr 4943  df-id 5216  df-eprel 5221  df-po 5229  df-so 5230  df-fr 5267  df-se 5268  df-we 5269  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-pred 5890  df-ord 5936  df-on 5937  df-lim 5938  df-suc 5939  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-isom 6107  df-riota 6832  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-om 7293  df-1st 7395  df-2nd 7396  df-wrecs 7639  df-recs 7701  df-rdg 7739  df-1o 7793  df-2o 7794  df-oadd 7797  df-er 7976  df-map 8091  df-pm 8092  df-en 8190  df-dom 8191  df-sdom 8192  df-fin 8193  df-card 9045  df-ac 9219  df-cda 9272  df-pnf 10358  df-mnf 10359  df-xr 10360  df-ltxr 10361  df-le 10362  df-sub 10550  df-neg 10551  df-nn 11303  df-2 11360  df-3 11361  df-n0 11556  df-xnn0 11626  df-z 11640  df-uz 11901  df-fz 12546  df-fzo 12686  df-hash 13334  df-word 13506  df-concat 13508  df-s1 13509  df-s2 13813  df-s3 13814  df-edg 26150  df-uhgr 26163  df-upgr 26187  df-umgr 26188  df-usgr 26257  df-wlks 26719  df-wwlks 26947  df-wwlksn 26948  df-wwlksnon 26949  df-frgr 27428
This theorem is referenced by:  frgr2wsp1  27501
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