Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frgr2wwlkeqm Structured version   Visualization version   GIF version

Theorem frgr2wwlkeqm 41496
 Description: If there is a (simple) path of length 2 from one vertex to another vertex and a (simple) path of length 2 from the other vertex back to the first vertex in a friendship graph, then the middle vertex is the same. (Contributed by Alexander van der Vekens, 20-Feb-2018.) (Revised by AV, 13-May-2021.)
Assertion
Ref Expression
frgr2wwlkeqm ((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → 𝑄 = 𝑃))

Proof of Theorem frgr2wwlkeqm
Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . . . . . 9 (Vtx‘𝐺) = (Vtx‘𝐺)
21wwlks2onv 41158 . . . . . . . 8 ((𝑃𝑋 ∧ ⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
32ex 449 . . . . . . 7 (𝑃𝑋 → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))))
43adantr 480 . . . . . 6 ((𝑃𝑋𝑄𝑌) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))))
543ad2ant3 1077 . . . . 5 ((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))))
61wwlks2onv 41158 . . . . . . . 8 ((𝑄𝑌 ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)))
76ex 449 . . . . . . 7 (𝑄𝑌 → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))))
87adantl 481 . . . . . 6 ((𝑃𝑋𝑄𝑌) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))))
983ad2ant3 1077 . . . . 5 ((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))))
105, 9anim12d 584 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)))))
1110imp 444 . . 3 (((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) ∧ (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴))) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))))
12 frgrusgr 41432 . . . . . . . . 9 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
13 usgrumgr 40409 . . . . . . . . 9 (𝐺 ∈ USGraph → 𝐺 ∈ UMGraph )
1412, 13syl 17 . . . . . . . 8 (𝐺 ∈ FriendGraph → 𝐺 ∈ UMGraph )
15143ad2ant1 1075 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → 𝐺 ∈ UMGraph )
16 simpl 472 . . . . . . 7 (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
17 eqid 2610 . . . . . . . 8 (Edg‘𝐺) = (Edg‘𝐺)
181, 17umgrwwlks2on 41161 . . . . . . 7 ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺))) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺))))
1915, 16, 18syl2anr 494 . . . . . 6 ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ↔ ({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺))))
20 simpr 476 . . . . . . 7 (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) → (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)))
211, 17umgrwwlks2on 41161 . . . . . . 7 ((𝐺 ∈ UMGraph ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) ↔ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺))))
2215, 20, 21syl2anr 494 . . . . . 6 ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → (⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴) ↔ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺))))
2319, 22anbi12d 743 . . . . 5 ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) ↔ (({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺)))))
24 simp1 1054 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → 𝐺 ∈ FriendGraph )
2524adantl 481 . . . . . . 7 ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → 𝐺 ∈ FriendGraph )
26 simpll1 1093 . . . . . . . 8 ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → 𝐴 ∈ (Vtx‘𝐺))
27 simpll3 1095 . . . . . . . 8 ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → 𝐵 ∈ (Vtx‘𝐺))
28 simpr2 1061 . . . . . . . 8 ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → 𝐴𝐵)
2926, 27, 283jca 1235 . . . . . . 7 ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐴𝐵))
301, 17frcond2 41439 . . . . . . 7 (𝐺 ∈ FriendGraph → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐴𝐵) → ∃!𝑝 ∈ (Vtx‘𝐺)({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺))))
3125, 29, 30sylc 63 . . . . . 6 ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → ∃!𝑝 ∈ (Vtx‘𝐺)({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)))
32 preq2 4213 . . . . . . . . . 10 (𝑝 = 𝑞 → {𝐴, 𝑝} = {𝐴, 𝑞})
3332eleq1d 2672 . . . . . . . . 9 (𝑝 = 𝑞 → ({𝐴, 𝑝} ∈ (Edg‘𝐺) ↔ {𝐴, 𝑞} ∈ (Edg‘𝐺)))
34 preq1 4212 . . . . . . . . . 10 (𝑝 = 𝑞 → {𝑝, 𝐵} = {𝑞, 𝐵})
3534eleq1d 2672 . . . . . . . . 9 (𝑝 = 𝑞 → ({𝑝, 𝐵} ∈ (Edg‘𝐺) ↔ {𝑞, 𝐵} ∈ (Edg‘𝐺)))
3633, 35anbi12d 743 . . . . . . . 8 (𝑝 = 𝑞 → (({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) ↔ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))))
3736reu4 3367 . . . . . . 7 (∃!𝑝 ∈ (Vtx‘𝐺)({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) ↔ (∃𝑝 ∈ (Vtx‘𝐺)({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) ∧ ∀𝑝 ∈ (Vtx‘𝐺)∀𝑞 ∈ (Vtx‘𝐺)((({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))) → 𝑝 = 𝑞)))
38 preq2 4213 . . . . . . . . . . . . . . . . . . 19 (𝑝 = 𝑃 → {𝐴, 𝑝} = {𝐴, 𝑃})
3938eleq1d 2672 . . . . . . . . . . . . . . . . . 18 (𝑝 = 𝑃 → ({𝐴, 𝑝} ∈ (Edg‘𝐺) ↔ {𝐴, 𝑃} ∈ (Edg‘𝐺)))
40 preq1 4212 . . . . . . . . . . . . . . . . . . 19 (𝑝 = 𝑃 → {𝑝, 𝐵} = {𝑃, 𝐵})
4140eleq1d 2672 . . . . . . . . . . . . . . . . . 18 (𝑝 = 𝑃 → ({𝑝, 𝐵} ∈ (Edg‘𝐺) ↔ {𝑃, 𝐵} ∈ (Edg‘𝐺)))
4239, 41anbi12d 743 . . . . . . . . . . . . . . . . 17 (𝑝 = 𝑃 → (({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) ↔ ({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺))))
4342anbi1d 737 . . . . . . . . . . . . . . . 16 (𝑝 = 𝑃 → ((({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))) ↔ (({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺)))))
44 eqeq1 2614 . . . . . . . . . . . . . . . 16 (𝑝 = 𝑃 → (𝑝 = 𝑞𝑃 = 𝑞))
4543, 44imbi12d 333 . . . . . . . . . . . . . . 15 (𝑝 = 𝑃 → (((({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))) → 𝑝 = 𝑞) ↔ ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))) → 𝑃 = 𝑞)))
46 preq2 4213 . . . . . . . . . . . . . . . . . . . 20 (𝑞 = 𝑄 → {𝐴, 𝑞} = {𝐴, 𝑄})
4746eleq1d 2672 . . . . . . . . . . . . . . . . . . 19 (𝑞 = 𝑄 → ({𝐴, 𝑞} ∈ (Edg‘𝐺) ↔ {𝐴, 𝑄} ∈ (Edg‘𝐺)))
48 preq1 4212 . . . . . . . . . . . . . . . . . . . 20 (𝑞 = 𝑄 → {𝑞, 𝐵} = {𝑄, 𝐵})
4948eleq1d 2672 . . . . . . . . . . . . . . . . . . 19 (𝑞 = 𝑄 → ({𝑞, 𝐵} ∈ (Edg‘𝐺) ↔ {𝑄, 𝐵} ∈ (Edg‘𝐺)))
5047, 49anbi12d 743 . . . . . . . . . . . . . . . . . 18 (𝑞 = 𝑄 → (({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺)) ↔ ({𝐴, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐵} ∈ (Edg‘𝐺))))
5150anbi2d 736 . . . . . . . . . . . . . . . . 17 (𝑞 = 𝑄 → ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))) ↔ (({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐵} ∈ (Edg‘𝐺)))))
52 eqeq2 2621 . . . . . . . . . . . . . . . . 17 (𝑞 = 𝑄 → (𝑃 = 𝑞𝑃 = 𝑄))
5351, 52imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑞 = 𝑄 → (((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))) → 𝑃 = 𝑞) ↔ ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐵} ∈ (Edg‘𝐺))) → 𝑃 = 𝑄)))
54 prcom 4211 . . . . . . . . . . . . . . . . . . . 20 {𝐴, 𝑄} = {𝑄, 𝐴}
5554eleq1i 2679 . . . . . . . . . . . . . . . . . . 19 ({𝐴, 𝑄} ∈ (Edg‘𝐺) ↔ {𝑄, 𝐴} ∈ (Edg‘𝐺))
56 prcom 4211 . . . . . . . . . . . . . . . . . . . 20 {𝑄, 𝐵} = {𝐵, 𝑄}
5756eleq1i 2679 . . . . . . . . . . . . . . . . . . 19 ({𝑄, 𝐵} ∈ (Edg‘𝐺) ↔ {𝐵, 𝑄} ∈ (Edg‘𝐺))
5855, 57anbi12ci 730 . . . . . . . . . . . . . . . . . 18 (({𝐴, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐵} ∈ (Edg‘𝐺)) ↔ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺)))
5958anbi2i 726 . . . . . . . . . . . . . . . . 17 ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐵} ∈ (Edg‘𝐺))) ↔ (({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺))))
60 eqcom 2617 . . . . . . . . . . . . . . . . 17 (𝑃 = 𝑄𝑄 = 𝑃)
6159, 60imbi12i 339 . . . . . . . . . . . . . . . 16 (((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐵} ∈ (Edg‘𝐺))) → 𝑃 = 𝑄) ↔ ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺))) → 𝑄 = 𝑃))
6253, 61syl6bb 275 . . . . . . . . . . . . . . 15 (𝑞 = 𝑄 → (((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))) → 𝑃 = 𝑞) ↔ ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺))) → 𝑄 = 𝑃)))
6345, 62rspc2v 3293 . . . . . . . . . . . . . 14 ((𝑃 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺)) → (∀𝑝 ∈ (Vtx‘𝐺)∀𝑞 ∈ (Vtx‘𝐺)((({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))) → 𝑝 = 𝑞) → ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺))) → 𝑄 = 𝑃)))
6463ex 449 . . . . . . . . . . . . 13 (𝑃 ∈ (Vtx‘𝐺) → (𝑄 ∈ (Vtx‘𝐺) → (∀𝑝 ∈ (Vtx‘𝐺)∀𝑞 ∈ (Vtx‘𝐺)((({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))) → 𝑝 = 𝑞) → ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺))) → 𝑄 = 𝑃))))
65643ad2ant2 1076 . . . . . . . . . . . 12 ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (𝑄 ∈ (Vtx‘𝐺) → (∀𝑝 ∈ (Vtx‘𝐺)∀𝑞 ∈ (Vtx‘𝐺)((({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))) → 𝑝 = 𝑞) → ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺))) → 𝑄 = 𝑃))))
6665com12 32 . . . . . . . . . . 11 (𝑄 ∈ (Vtx‘𝐺) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (∀𝑝 ∈ (Vtx‘𝐺)∀𝑞 ∈ (Vtx‘𝐺)((({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))) → 𝑝 = 𝑞) → ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺))) → 𝑄 = 𝑃))))
67663ad2ant2 1076 . . . . . . . . . 10 ((𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺)) → ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (∀𝑝 ∈ (Vtx‘𝐺)∀𝑞 ∈ (Vtx‘𝐺)((({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))) → 𝑝 = 𝑞) → ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺))) → 𝑄 = 𝑃))))
6867impcom 445 . . . . . . . . 9 (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) → (∀𝑝 ∈ (Vtx‘𝐺)∀𝑞 ∈ (Vtx‘𝐺)((({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))) → 𝑝 = 𝑞) → ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺))) → 𝑄 = 𝑃)))
6968adantr 480 . . . . . . . 8 ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → (∀𝑝 ∈ (Vtx‘𝐺)∀𝑞 ∈ (Vtx‘𝐺)((({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))) → 𝑝 = 𝑞) → ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺))) → 𝑄 = 𝑃)))
7069com12 32 . . . . . . 7 (∀𝑝 ∈ (Vtx‘𝐺)∀𝑞 ∈ (Vtx‘𝐺)((({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐴, 𝑞} ∈ (Edg‘𝐺) ∧ {𝑞, 𝐵} ∈ (Edg‘𝐺))) → 𝑝 = 𝑞) → ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺))) → 𝑄 = 𝑃)))
7137, 70simplbiim 657 . . . . . 6 (∃!𝑝 ∈ (Vtx‘𝐺)({𝐴, 𝑝} ∈ (Edg‘𝐺) ∧ {𝑝, 𝐵} ∈ (Edg‘𝐺)) → ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺))) → 𝑄 = 𝑃)))
7231, 71mpcom 37 . . . . 5 ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → ((({𝐴, 𝑃} ∈ (Edg‘𝐺) ∧ {𝑃, 𝐵} ∈ (Edg‘𝐺)) ∧ ({𝐵, 𝑄} ∈ (Edg‘𝐺) ∧ {𝑄, 𝐴} ∈ (Edg‘𝐺))) → 𝑄 = 𝑃))
7323, 72sylbid 229 . . . 4 ((((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) ∧ (𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌))) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → 𝑄 = 𝑃))
7473expimpd 627 . . 3 (((𝐴 ∈ (Vtx‘𝐺) ∧ 𝑃 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐵 ∈ (Vtx‘𝐺) ∧ 𝑄 ∈ (Vtx‘𝐺) ∧ 𝐴 ∈ (Vtx‘𝐺))) → (((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) ∧ (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴))) → 𝑄 = 𝑃))
7511, 74mpcom 37 . 2 (((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) ∧ (⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴))) → 𝑄 = 𝑃)
7675ex 449 1 ((𝐺 ∈ FriendGraph ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → 𝑄 = 𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  {cpr 4127  ‘cfv 5804  (class class class)co 6549  2c2 10947  ⟨“cs3 13438  Vtxcvtx 25673   UMGraph cumgr 25748  Edgcedga 25792   USGraph cusgr 40379   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: (None)
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