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Theorem frgr2wwlk1 27046
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.)
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 27045 . . 3 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅)
31elwwlks2ons3 26711 . . . . . . . . . . 11 ((𝐺 ∈ FriendGraph ∧ 𝐴𝑉𝐵𝑉) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
433expb 1263 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉)) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
543adant3 1079 . . . . . . . . 9 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
61elwwlks2ons3 26711 . . . . . . . . . . 11 ((𝐺 ∈ FriendGraph ∧ 𝐴𝑉𝐵𝑉) → (𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
763expb 1263 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉)) → (𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
873adant3 1079 . . . . . . . . 9 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
95, 8anbi12d 746 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((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)))))
101frgr2wwlkeu 27044 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑥𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
11 eqidd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑦𝐴 = 𝐴)
12 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑦𝑥 = 𝑦)
13 eqidd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑦𝐵 = 𝐵)
1411, 12, 13s3eqd 13541 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑦 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑦𝐵”⟩)
1514eleq1d 2688 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑦 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
1615reu4 3387 . . . . . . . . . . . . . . . . . . . . . . 23 (∃!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)))
17 eqidd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = 𝑐𝐴 = 𝐴)
18 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = 𝑐𝑥 = 𝑐)
19 eqidd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = 𝑐𝐵 = 𝐵)
2017, 18, 19s3eqd 13541 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = 𝑐 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩)
2120eleq1d 2688 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 𝑐 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
2221anbi1d 740 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = 𝑐 → ((⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ↔ (⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
23 equequ1 1954 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 𝑐 → (𝑥 = 𝑦𝑐 = 𝑦))
24 equcom 1947 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 = 𝑦𝑦 = 𝑐)
2523, 24syl6bb 276 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = 𝑐 → (𝑥 = 𝑦𝑦 = 𝑐))
2622, 25imbi12d 334 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑐 → (((⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑥 = 𝑦) ↔ ((⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑦 = 𝑐)))
27 eqidd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = 𝑑𝐴 = 𝐴)
28 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = 𝑑𝑦 = 𝑑)
29 eqidd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = 𝑑𝐵 = 𝐵)
3027, 28, 29s3eqd 13541 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = 𝑑 → ⟨“𝐴𝑦𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
3130eleq1d 2688 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = 𝑑 → (⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
3231anbi2d 739 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = 𝑑 → ((⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ↔ (⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
33 ancom 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ↔ (⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
3432, 33syl6bb 276 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑑 → ((⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ↔ (⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
35 equequ1 1954 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑑 → (𝑦 = 𝑐𝑑 = 𝑐))
3634, 35imbi12d 334 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑑 → (((⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑦 = 𝑐) ↔ ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑑 = 𝑐)))
3726, 36rspc2va 3312 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑐𝑉𝑑𝑉) ∧ ∀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))
38 eqidd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 = 𝑐𝐴 = 𝐴)
39 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 = 𝑐𝑑 = 𝑐)
40 eqidd 2627 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 = 𝑐𝐵 = 𝐵)
4138, 39, 40s3eqd 13541 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = 𝑐 → ⟨“𝐴𝑑𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩)
4237, 41syl6 35 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑐𝑉𝑑𝑉) ∧ ∀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)) → ⟨“𝐴𝑑𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩))
4342expcom 451 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑥𝑉𝑦𝑉 ((⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑥 = 𝑦) → ((𝑐𝑉𝑑𝑉) → ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ⟨“𝐴𝑑𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩)))
4416, 43simplbiim 658 . . . . . . . . . . . . . . . . . . . . . 22 (∃!𝑥𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → ((𝑐𝑉𝑑𝑉) → ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ⟨“𝐴𝑑𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩)))
4510, 44syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑐𝑉𝑑𝑉) → ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ⟨“𝐴𝑑𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩)))
4645impl 649 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ⟨“𝐴𝑑𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩))
47 eleq1 2692 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
48 eleq1 2692 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ → (𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
4947, 48bi2anan9 916 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ 𝑡 = ⟨“𝐴𝑐𝐵”⟩) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ↔ (⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
50 eqeq12 2639 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ 𝑡 = ⟨“𝐴𝑐𝐵”⟩) → (𝑤 = 𝑡 ↔ ⟨“𝐴𝑑𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩))
5149, 50imbi12d 334 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ 𝑡 = ⟨“𝐴𝑐𝐵”⟩) → (((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡) ↔ ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ⟨“𝐴𝑑𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩)))
5246, 51syl5ibr 236 . . . . . . . . . . . . . . . . . . 19 ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ 𝑡 = ⟨“𝐴𝑐𝐵”⟩) → ((((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
5352ex 450 . . . . . . . . . . . . . . . . . 18 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ → (𝑡 = ⟨“𝐴𝑐𝐵”⟩ → ((((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))))
5453com23 86 . . . . . . . . . . . . . . . . 17 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ → ((((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → (𝑡 = ⟨“𝐴𝑐𝐵”⟩ → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))))
5554adantr 481 . . . . . . . . . . . . . . . 16 ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → (𝑡 = ⟨“𝐴𝑐𝐵”⟩ → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))))
5655com12 32 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (𝑡 = ⟨“𝐴𝑐𝐵”⟩ → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))))
5756rexlimdva 3029 . . . . . . . . . . . . . 14 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) → (∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (𝑡 = ⟨“𝐴𝑐𝐵”⟩ → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))))
5857com23 86 . . . . . . . . . . . . 13 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) → (𝑡 = ⟨“𝐴𝑐𝐵”⟩ → (∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))))
5958adantrd 484 . . . . . . . . . . . 12 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) → ((𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))))
6059rexlimdva 3029 . . . . . . . . . . 11 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))))
6160com13 88 . . . . . . . . . 10 (∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))))
6261imp 445 . . . . . . . . 9 ((∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ∧ ∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))) → ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
6362com12 32 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) ∧ ∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
649, 63sylbid 230 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡)))
6564pm2.43d 53 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))
6665alrimivv 1858 . . . . 5 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∀𝑤𝑡((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))
67 eleq1 2692 . . . . . 6 (𝑤 = 𝑡 → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
6867mo4 2521 . . . . 5 (∃*𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∀𝑤𝑡((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))
6966, 68sylibr 224 . . . 4 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃*𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
70 n0moeu 3918 . . . 4 ((𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅ → (∃*𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
7169, 70syl5ib 234 . . 3 ((𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅ → ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
722, 71mpcom 38 . 2 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
73 ovex 6633 . . 3 (𝐴(2 WWalksNOn 𝐺)𝐵) ∈ V
74 euhash1 13145 . . 3 ((𝐴(2 WWalksNOn 𝐺)𝐵) ∈ V → ((#‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1 ↔ ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
7573, 74mp1i 13 . 2 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((#‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1 ↔ ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
7672, 75mpbird 247 1 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (#‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1992  ∃!weu 2474  ∃*wmo 2475  wne 2796  wral 2912  wrex 2913  ∃!wreu 2914  Vcvv 3191  c0 3896  cfv 5850  (class class class)co 6605  1c1 9882  2c2 11015  #chash 13054  ⟨“cs3 13519  Vtxcvtx 25769   WWalksNOn cwwlksnon 26582   FriendGraph cfrgr 26980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-ac2 9230  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-ac 8884  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-fzo 12404  df-hash 13055  df-word 13233  df-concat 13235  df-s1 13236  df-s2 13525  df-s3 13526  df-edg 25835  df-uhgr 25844  df-upgr 25868  df-umgr 25869  df-usgr 25934  df-wlks 26359  df-wwlks 26585  df-wwlksn 26586  df-wwlksnon 26587  df-frgr 26981
This theorem is referenced by:  frgr2wsp1  27047
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