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Theorem frg2wot1 26584
 Description: In a friendship graph, there is exactly one walk of length 2 between two different vertices as ordered triple. (Contributed by Alexander van der Vekens, 19-Feb-2018.)
Assertion
Ref Expression
frg2wot1 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (#‘(𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) = 1)

Proof of Theorem frg2wot1
Dummy variables 𝑐 𝑑 𝑓 𝑝 𝑡 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frg2wotn0 26583 . . 3 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ≠ ∅)
2 frisusgra 26519 . . . . . . . . . . . . 13 (𝑉 FriendGrph 𝐸𝑉 USGrph 𝐸)
3 usgrav 25867 . . . . . . . . . . . . 13 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
42, 3syl 17 . . . . . . . . . . . 12 (𝑉 FriendGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
54anim1i 590 . . . . . . . . . . 11 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))
653adant3 1074 . . . . . . . . . 10 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)))
7 el2wlkonot 26396 . . . . . . . . . 10 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ∃𝑑𝑉 (𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2))))))
86, 7syl 17 . . . . . . . . 9 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ∃𝑑𝑉 (𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2))))))
9 el2wlkonot 26396 . . . . . . . . . 10 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ∃𝑐𝑉 (𝑡 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2))))))
106, 9syl 17 . . . . . . . . 9 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ∃𝑐𝑉 (𝑡 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2))))))
118, 10anbi12d 743 . . . . . . . 8 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) ↔ (∃𝑑𝑉 (𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) ∧ ∃𝑐𝑉 (𝑡 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))))))
12 frg2woteu 26582 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑥𝑉𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵))
13 oteq2 4350 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = 𝑦 → ⟨𝐴, 𝑥, 𝐵⟩ = ⟨𝐴, 𝑦, 𝐵⟩)
1413eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑦 → (⟨𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)))
1514reu4 3367 . . . . . . . . . . . . . . . . . . . . . . 23 (∃!𝑥𝑉𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ (∃𝑥𝑉𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ∀𝑥𝑉𝑦𝑉 ((⟨𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑥 = 𝑦)))
16 oteq2 4350 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥 = 𝑐 → ⟨𝐴, 𝑥, 𝐵⟩ = ⟨𝐴, 𝑐, 𝐵⟩)
1716eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥 = 𝑐 → (⟨𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)))
1817anbi1d 737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = 𝑐 → ((⟨𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) ↔ (⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵))))
19 ancom 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) ↔ (⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)))
2018, 19syl6bb 275 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = 𝑐 → ((⟨𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) ↔ (⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵))))
21 equequ1 1939 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑥 = 𝑐 → (𝑥 = 𝑦𝑐 = 𝑦))
22 equcom 1932 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑐 = 𝑦𝑦 = 𝑐)
2321, 22syl6bb 275 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = 𝑐 → (𝑥 = 𝑦𝑦 = 𝑐))
2420, 23imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 𝑐 → (((⟨𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑥 = 𝑦) ↔ ((⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑦 = 𝑐)))
25 oteq2 4350 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = 𝑑 → ⟨𝐴, 𝑦, 𝐵⟩ = ⟨𝐴, 𝑑, 𝐵⟩)
2625eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = 𝑑 → (⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ⟨𝐴, 𝑑, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)))
2726anbi1d 737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = 𝑑 → ((⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) ↔ (⟨𝐴, 𝑑, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵))))
28 equequ1 1939 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = 𝑑 → (𝑦 = 𝑐𝑑 = 𝑐))
2927, 28imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = 𝑑 → (((⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑦 = 𝑐) ↔ ((⟨𝐴, 𝑑, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑑 = 𝑐)))
3024, 29rspc2va 3294 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑐𝑉𝑑𝑉) ∧ ∀𝑥𝑉𝑦𝑉 ((⟨𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑥 = 𝑦)) → ((⟨𝐴, 𝑑, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑑 = 𝑐))
31 oteq2 4350 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑑 = 𝑐 → ⟨𝐴, 𝑑, 𝐵⟩ = ⟨𝐴, 𝑐, 𝐵⟩)
3230, 31syl6 34 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑐𝑉𝑑𝑉) ∧ ∀𝑥𝑉𝑦𝑉 ((⟨𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑥 = 𝑦)) → ((⟨𝐴, 𝑑, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → ⟨𝐴, 𝑑, 𝐵⟩ = ⟨𝐴, 𝑐, 𝐵⟩))
3332ex 449 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑐𝑉𝑑𝑉) → (∀𝑥𝑉𝑦𝑉 ((⟨𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑥 = 𝑦) → ((⟨𝐴, 𝑑, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → ⟨𝐴, 𝑑, 𝐵⟩ = ⟨𝐴, 𝑐, 𝐵⟩)))
3433com12 32 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑥𝑉𝑦𝑉 ((⟨𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑥 = 𝑦) → ((𝑐𝑉𝑑𝑉) → ((⟨𝐴, 𝑑, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → ⟨𝐴, 𝑑, 𝐵⟩ = ⟨𝐴, 𝑐, 𝐵⟩)))
3534adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((∃𝑥𝑉𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ∀𝑥𝑉𝑦𝑉 ((⟨𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑦, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑥 = 𝑦)) → ((𝑐𝑉𝑑𝑉) → ((⟨𝐴, 𝑑, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → ⟨𝐴, 𝑑, 𝐵⟩ = ⟨𝐴, 𝑐, 𝐵⟩)))
3615, 35sylbi 206 . . . . . . . . . . . . . . . . . . . . . 22 (∃!𝑥𝑉𝐴, 𝑥, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) → ((𝑐𝑉𝑑𝑉) → ((⟨𝐴, 𝑑, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → ⟨𝐴, 𝑑, 𝐵⟩ = ⟨𝐴, 𝑐, 𝐵⟩)))
3712, 36syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑐𝑉𝑑𝑉) → ((⟨𝐴, 𝑑, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → ⟨𝐴, 𝑑, 𝐵⟩ = ⟨𝐴, 𝑐, 𝐵⟩)))
3837impl 648 . . . . . . . . . . . . . . . . . . . 20 ((((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → ((⟨𝐴, 𝑑, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → ⟨𝐴, 𝑑, 𝐵⟩ = ⟨𝐴, 𝑐, 𝐵⟩))
39 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ → (𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ⟨𝐴, 𝑑, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)))
40 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . 22 (𝑡 = ⟨𝐴, 𝑐, 𝐵⟩ → (𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)))
4139, 40bi2anan9 913 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ 𝑡 = ⟨𝐴, 𝑐, 𝐵⟩) → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) ↔ (⟨𝐴, 𝑑, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵))))
42 eqeq12 2623 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ 𝑡 = ⟨𝐴, 𝑐, 𝐵⟩) → (𝑤 = 𝑡 ↔ ⟨𝐴, 𝑑, 𝐵⟩ = ⟨𝐴, 𝑐, 𝐵⟩))
4341, 42imbi12d 333 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ 𝑡 = ⟨𝐴, 𝑐, 𝐵⟩) → (((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡) ↔ ((⟨𝐴, 𝑑, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → ⟨𝐴, 𝑑, 𝐵⟩ = ⟨𝐴, 𝑐, 𝐵⟩)))
4438, 43syl5ibr 235 . . . . . . . . . . . . . . . . . . 19 ((𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ 𝑡 = ⟨𝐴, 𝑐, 𝐵⟩) → ((((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡)))
4544ex 449 . . . . . . . . . . . . . . . . . 18 (𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ → (𝑡 = ⟨𝐴, 𝑐, 𝐵⟩ → ((((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡))))
4645com23 84 . . . . . . . . . . . . . . . . 17 (𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ → ((((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → (𝑡 = ⟨𝐴, 𝑐, 𝐵⟩ → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡))))
4746adantr 480 . . . . . . . . . . . . . . . 16 ((𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → ((((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → (𝑡 = ⟨𝐴, 𝑐, 𝐵⟩ → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡))))
4847com12 32 . . . . . . . . . . . . . . 15 ((((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) ∧ 𝑑𝑉) → ((𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → (𝑡 = ⟨𝐴, 𝑐, 𝐵⟩ → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡))))
4948rexlimdva 3013 . . . . . . . . . . . . . 14 (((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) → (∃𝑑𝑉 (𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → (𝑡 = ⟨𝐴, 𝑐, 𝐵⟩ → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡))))
5049com23 84 . . . . . . . . . . . . 13 (((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) → (𝑡 = ⟨𝐴, 𝑐, 𝐵⟩ → (∃𝑑𝑉 (𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡))))
5150adantrd 483 . . . . . . . . . . . 12 (((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) ∧ 𝑐𝑉) → ((𝑡 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → (∃𝑑𝑉 (𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡))))
5251rexlimdva 3013 . . . . . . . . . . 11 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (∃𝑐𝑉 (𝑡 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → (∃𝑑𝑉 (𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡))))
5352com13 86 . . . . . . . . . 10 (∃𝑑𝑉 (𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → (∃𝑐𝑉 (𝑡 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) → ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡))))
5453imp 444 . . . . . . . . 9 ((∃𝑑𝑉 (𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) ∧ ∃𝑐𝑉 (𝑡 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2))))) → ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡)))
5554com12 32 . . . . . . . 8 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((∃𝑑𝑉 (𝑤 = ⟨𝐴, 𝑑, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑑 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2)))) ∧ ∃𝑐𝑉 (𝑡 = ⟨𝐴, 𝑐, 𝐵⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑐 = (𝑝‘1) ∧ 𝐵 = (𝑝‘2))))) → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡)))
5611, 55sylbid 229 . . . . . . 7 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡)))
5756pm2.43d 51 . . . . . 6 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡))
5857alrimivv 1843 . . . . 5 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∀𝑤𝑡((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡))
59 eleq1 2676 . . . . . 6 (𝑤 = 𝑡 → (𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)))
6059mo4 2505 . . . . 5 (∃*𝑤 𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ∀𝑤𝑡((𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑡 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) → 𝑤 = 𝑡))
6158, 60sylibr 223 . . . 4 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃*𝑤 𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵))
62 n0moeu 3893 . . . 4 ((𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ≠ ∅ → (∃*𝑤 𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ↔ ∃!𝑤 𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)))
6361, 62syl5ib 233 . . 3 ((𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ≠ ∅ → ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑤 𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)))
641, 63mpcom 37 . 2 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑤 𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵))
65 ovex 6577 . . 3 (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∈ V
66 euhash1 13069 . . 3 ((𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∈ V → ((#‘(𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) = 1 ↔ ∃!𝑤 𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)))
6765, 66mp1i 13 . 2 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((#‘(𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) = 1 ↔ ∃!𝑤 𝑤 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵)))
6864, 67mpbird 246 1 ((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (#‘(𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) = 1)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  ∃*wmo 2459   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  Vcvv 3173  ∅c0 3874  ⟨cotp 4133   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  #chash 12979   USGrph cusg 25859   Walks cwalk 26026   2WalksOnOt c2wlkonot 26382   FriendGrph cfrgra 26515 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-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-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-ot 4134  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-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-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-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-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-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-wlkon 26042  df-2wlkonot 26385  df-frgra 26516 This theorem is referenced by:  frg2spot1  26585
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