Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frgr2wwlk1 Structured version   Visualization version   GIF version

Theorem frgr2wwlk1 41494
 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 41493 . . 3 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅)
31elwwlks2ons3 41159 . . . . . . . . . . 11 ((𝐺 ∈ FriendGraph ∧ 𝐴𝑉𝐵𝑉) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
433expb 1258 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉)) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
543adant3 1074 . . . . . . . . 9 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑑𝑉 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
61elwwlks2ons3 41159 . . . . . . . . . . 11 ((𝐺 ∈ FriendGraph ∧ 𝐴𝑉𝐵𝑉) → (𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
763expb 1258 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉)) → (𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
873adant3 1074 . . . . . . . . 9 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃𝑐𝑉 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))))
95, 8anbi12d 743 . . . . . . . 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 41492 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑥𝑉 ⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
11 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑦𝐴 = 𝐴)
12 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑦𝑥 = 𝑦)
13 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑦𝐵 = 𝐵)
1411, 12, 13s3eqd 13460 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑦 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑦𝐵”⟩)
1514eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑦 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
1615reu4 3367 . . . . . . . . . . . . . . . . . . . . . . 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 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = 𝑐𝐴 = 𝐴)
18 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = 𝑐𝑥 = 𝑐)
19 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = 𝑐𝐵 = 𝐵)
2017, 18, 19s3eqd 13460 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = 𝑐 → ⟨“𝐴𝑥𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩)
2120eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 𝑐 → (⟨“𝐴𝑥𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
2221anbi1d 737 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1939 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 𝑐 → (𝑥 = 𝑦𝑐 = 𝑦))
24 equcom 1932 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 = 𝑦𝑦 = 𝑐)
2523, 24syl6bb 275 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 = 𝑐 → (𝑥 = 𝑦𝑦 = 𝑐))
2622, 25imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = 𝑑𝐴 = 𝐴)
28 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = 𝑑𝑦 = 𝑑)
29 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = 𝑑𝐵 = 𝐵)
3027, 28, 29s3eqd 13460 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = 𝑑 → ⟨“𝐴𝑦𝐵”⟩ = ⟨“𝐴𝑑𝐵”⟩)
3130eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = 𝑑 → (⟨“𝐴𝑦𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
3231anbi2d 736 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 275 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1939 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = 𝑑 → (𝑦 = 𝑐𝑑 = 𝑐))
3634, 35imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3294 . . . . . . . . . . . . . . . . . . . . . . . . 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 2611 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 = 𝑐𝐴 = 𝐴)
39 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 = 𝑐𝑑 = 𝑐)
40 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑑 = 𝑐𝐵 = 𝐵)
4138, 39, 40s3eqd 13460 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑑 = 𝑐 → ⟨“𝐴𝑑𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩)
4237, 41syl6 34 . . . . . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . . . . . . 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 657 . . . . . . . . . . . . . . . . . . . . . 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 648 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → ((⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → ⟨“𝐴𝑑𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩))
47 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = ⟨“𝐴𝑑𝐵”⟩ → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑑𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
48 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = ⟨“𝐴𝑐𝐵”⟩ → (𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
4947, 48bi2anan9 913 . . . . . . . . . . . . . . . . . . . . 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 2623 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 = ⟨“𝐴𝑑𝐵”⟩ ∧ 𝑡 = ⟨“𝐴𝑐𝐵”⟩) → (𝑤 = 𝑡 ↔ ⟨“𝐴𝑑𝐵”⟩ = ⟨“𝐴𝑐𝐵”⟩))
5149, 50imbi12d 333 . . . . . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . 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 84 . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 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 3013 . . . . . . . . . . . . . 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 84 . . . . . . . . . . . . 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 483 . . . . . . . . . . . 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 3013 . . . . . . . . . . 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 86 . . . . . . . . . 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 444 . . . . . . . . 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 229 . . . . . . 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 51 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))
6665alrimivv 1843 . . . . 5 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∀𝑤𝑡((𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → 𝑤 = 𝑡))
67 eleq1 2676 . . . . . 6 (𝑤 = 𝑡 → (𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ 𝑡 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
6867mo4 2505 . . . . 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 223 . . . 4 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃*𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
70 n0moeu 3893 . . . 4 ((𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅ → (∃*𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
7169, 70syl5ib 233 . . 3 ((𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅ → ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)))
722, 71mpcom 37 . 2 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑤 𝑤 ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))
73 ovex 6577 . . 3 (𝐴(2 WWalksNOn 𝐺)𝐵) ∈ V
74 euhash1 13069 . . 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 246 1 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (#‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃!weu 2458  ∃*wmo 2459   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  Vcvv 3173  ∅c0 3874  ‘cfv 5804  (class class class)co 6549  1c1 9816  2c2 10947  #chash 12979  ⟨“cs3 13438  Vtxcvtx 25673   WWalksNOn cwwlksnon 41030   FriendGraph cfrgr 41428 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-ac2 9168  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-ifp 1007  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-rmo 2904  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-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-se 4998  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-isom 5813  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-2o 7448  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-ac 8822  df-cda 8873  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-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-uhgr 25724  df-upgr 25749  df-umgr 25750  df-edga 25793  df-usgr 40381  df-1wlks 40800  df-wlks 40801  df-wwlks 41033  df-wwlksn 41034  df-wwlksnon 41035  df-frgr 41429 This theorem is referenced by:  frgr2wsp1  41495
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