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Theorem clnbgr3stgrgrlic 47979
Description: If all (closed) neighborhoods of the vertices in two simple graphs with the same order induce a subgraph which is isomorphic to an 𝑁-star, then the two graphs are locally isomorphic. (Contributed by AV, 29-Sep-2025.)
Hypotheses
Ref Expression
clnbgr3stgrgrlic.n 𝑁 ∈ ℕ0
clnbgr3stgrgrlic.v 𝑉 = (Vtx‘𝐺)
clnbgr3stgrgrlic.w 𝑊 = (Vtx‘𝐻)
Assertion
Ref Expression
clnbgr3stgrgrlic (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉𝑊) ∧ ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐺𝑙𝑔𝑟 𝐻)
Distinct variable groups:   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑥,𝑁,𝑦   𝑥,𝑉   𝑥,𝑊,𝑦
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem clnbgr3stgrgrlic
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 clnbgr3stgrgrlic.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
21fvexi 6920 . . . . . . 7 𝑉 ∈ V
3 clnbgr3stgrgrlic.w . . . . . . . 8 𝑊 = (Vtx‘𝐻)
43fvexi 6920 . . . . . . 7 𝑊 ∈ V
52, 4pm3.2i 470 . . . . . 6 (𝑉 ∈ V ∧ 𝑊 ∈ V)
6 breng 8994 . . . . . 6 ((𝑉 ∈ V ∧ 𝑊 ∈ V) → (𝑉𝑊 ↔ ∃𝑓 𝑓:𝑉1-1-onto𝑊))
75, 6mp1i 13 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝑉𝑊 ↔ ∃𝑓 𝑓:𝑉1-1-onto𝑊))
8 usgruhgr 29203 . . . . . . . . . . . . . . . . . . . 20 (𝐻 ∈ USGraph → 𝐻 ∈ UHGraph)
98adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → 𝐻 ∈ UHGraph)
1093ad2ant1 1134 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐻 ∈ UHGraph)
113clnbgrssvtx 47818 . . . . . . . . . . . . . . . . . . 19 (𝐻 ClNeighbVtx (𝑓𝑥)) ⊆ 𝑊
1211a1i 11 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥𝑉) → (𝐻 ClNeighbVtx (𝑓𝑥)) ⊆ 𝑊)
133isubgruhgr 47854 . . . . . . . . . . . . . . . . . 18 ((𝐻 ∈ UHGraph ∧ (𝐻 ClNeighbVtx (𝑓𝑥)) ⊆ 𝑊) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))) ∈ UHGraph)
1410, 12, 13syl2an2r 685 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥𝑉) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))) ∈ UHGraph)
15 f1of 6848 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓:𝑉1-1-onto𝑊𝑓:𝑉𝑊)
16153ad2ant2 1135 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊𝑥𝑉) → 𝑓:𝑉𝑊)
17 simp3 1139 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊𝑥𝑉) → 𝑥𝑉)
1816, 17ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊𝑥𝑉) → (𝑓𝑥) ∈ 𝑊)
19 oveq2 7439 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = (𝑓𝑥) → (𝐻 ClNeighbVtx 𝑦) = (𝐻 ClNeighbVtx (𝑓𝑥)))
2019oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (𝑓𝑥) → (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) = (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))))
2120breq1d 5153 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (𝑓𝑥) → ((𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) ↔ (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))
2221rspcv 3618 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓𝑥) ∈ 𝑊 → (∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))
2318, 22syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊𝑥𝑉) → (∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))
24233exp 1120 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝑓:𝑉1-1-onto𝑊 → (𝑥𝑉 → (∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))))
2524com34 91 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝑓:𝑉1-1-onto𝑊 → (∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝑥𝑉 → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))))
26253imp1 1348 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥𝑉) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))) ≃𝑔𝑟 (StarGr‘𝑁))
27 gricsym 47890 . . . . . . . . . . . . . . . . 17 ((𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))) ∈ UHGraph → ((𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))) ≃𝑔𝑟 (StarGr‘𝑁) → (StarGr‘𝑁) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥)))))
2814, 26, 27sylc 65 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥𝑉) → (StarGr‘𝑁) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))))
2928anim1ci 616 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥𝑉) ∧ (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁)) → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ (StarGr‘𝑁) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥)))))
30 grictr 47892 . . . . . . . . . . . . . . 15 (((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ (StarGr‘𝑁) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥)))) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))))
3129, 30syl 17 . . . . . . . . . . . . . 14 (((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥𝑉) ∧ (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))))
3231ex 412 . . . . . . . . . . . . 13 ((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥𝑉) → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥)))))
3332ralimdva 3167 . . . . . . . . . . . 12 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → (∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) → ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥)))))
34333exp 1120 . . . . . . . . . . 11 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝑓:𝑉1-1-onto𝑊 → (∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) → ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥)))))))
3534com24 95 . . . . . . . . . 10 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) → (∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝑓:𝑉1-1-onto𝑊 → ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥)))))))
3635imp32 418 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ (∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁))) → (𝑓:𝑉1-1-onto𝑊 → ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥)))))
3736ancld 550 . . . . . . . 8 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ (∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁))) → (𝑓:𝑉1-1-onto𝑊 → (𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))))))
3837eximdv 1917 . . . . . . 7 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ (∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁))) → (∃𝑓 𝑓:𝑉1-1-onto𝑊 → ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))))))
3938ex 412 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → ((∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → (∃𝑓 𝑓:𝑉1-1-onto𝑊 → ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥)))))))
4039com23 86 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (∃𝑓 𝑓:𝑉1-1-onto𝑊 → ((∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥)))))))
417, 40sylbid 240 . . . 4 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝑉𝑊 → ((∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥)))))))
42413impia 1118 . . 3 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉𝑊) → ((∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))))))
43423impib 1117 . 2 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉𝑊) ∧ ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥)))))
441, 3dfgrlic2 47968 . . . 4 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝐺𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))))))
45443adant3 1133 . . 3 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉𝑊) → (𝐺𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))))))
46453ad2ant1 1134 . 2 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉𝑊) ∧ ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → (𝐺𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))))))
4743, 46mpbird 257 1 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉𝑊) ∧ ∀𝑥𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐺𝑙𝑔𝑟 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wral 3061  Vcvv 3480  wss 3951   class class class wbr 5143  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cen 8982  0cn0 12526  Vtxcvtx 29013  UHGraphcuhgr 29073  USGraphcusgr 29166   ClNeighbVtx cclnbgr 47805   ISubGr cisubgr 47846  𝑔𝑟 cgric 47862  StarGrcstgr 47918  𝑙𝑔𝑟 cgrlic 47944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-i2m1 11223  ax-1ne0 11224  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-2 12329  df-vtx 29015  df-iedg 29016  df-uhgr 29075  df-upgr 29099  df-uspgr 29167  df-usgr 29168  df-clnbgr 47806  df-isubgr 47847  df-grim 47864  df-gric 47867  df-grlim 47945  df-grlic 47948
This theorem is referenced by:  gpg5grlic  48047
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