Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clnbgr3stgrgrlic Structured version   Visualization version   GIF version

Theorem clnbgr3stgrgrlic 47914
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 8992 . . . . . 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 29217 . . . . . . . . . . . . . . . . . . . 20 (𝐻 ∈ USGraph → 𝐻 ∈ UHGraph)
98adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → 𝐻 ∈ UHGraph)
1093ad2ant1 1132 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑦𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐻 ∈ UHGraph)
113clnbgrssvtx 47755 . . . . . . . . . . . . . . . . . . 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 47791 . . . . . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊𝑥𝑉) → 𝑓:𝑉𝑊)
17 simp3 1137 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊𝑥𝑉) → 𝑥𝑉)
1816, 17ffvelcdmd 7104 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉1-1-onto𝑊𝑥𝑉) → (𝑓𝑥) ∈ 𝑊)
19 oveq2 7438 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = (𝑓𝑥) → (𝐻 ClNeighbVtx 𝑦) = (𝐻 ClNeighbVtx (𝑓𝑥)))
2019oveq2d 7446 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (𝑓𝑥) → (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) = (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))))
2120breq1d 5157 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (𝑓𝑥) → ((𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) ↔ (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))
2221rspcv 3617 . . . . . . . . . . . . . . . . . . . . 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 1118 . . . . . . . . . . . . . . . . . . 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 1346 . . . . . . . . . . . . . . . . 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 47827 . . . . . . . . . . . . . . . . 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 47829 . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . 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 1118 . . . . . . . . . . 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 1914 . . . . . . 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 1116 . . 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 1115 . 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 47903 . . . 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 1131 . . 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 1132 . 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 1086   = wceq 1536  wex 1775  wcel 2105  wral 3058  Vcvv 3477  wss 3962   class class class wbr 5147  wf 6558  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  cen 8980  0cn0 12523  Vtxcvtx 29027  UHGraphcuhgr 29087  USGraphcusgr 29180   ClNeighbVtx cclnbgr 47742   ISubGr cisubgr 47783  𝑔𝑟 cgric 47799  StarGrcstgr 47853  𝑙𝑔𝑟 cgrlic 47879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-i2m1 11220  ax-1ne0 11221  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-2 12326  df-vtx 29029  df-iedg 29030  df-uhgr 29089  df-upgr 29113  df-uspgr 29181  df-usgr 29182  df-clnbgr 47743  df-isubgr 47784  df-grim 47801  df-gric 47804  df-grlim 47880  df-grlic 47883
This theorem is referenced by:  gpg5grlic  47974
  Copyright terms: Public domain W3C validator