Theorem clnbgr3stgrgrlic 48514

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
clnbgr3stgrgrlim.n	⊢ 𝑁 ∈ ℕ0
clnbgr3stgrgrlim.v	⊢ 𝑉 = (Vtx‘𝐺)
clnbgr3stgrgrlim.w	⊢ 𝑊 = (Vtx‘𝐻)

Assertion

Ref	Expression
clnbgr3stgrgrlic	⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐺 ≃𝑙𝑔𝑟 𝐻)

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝐻,𝑦   𝑥,𝑁,𝑦   𝑥,𝑉   𝑥,𝑊,𝑦   𝑦,𝐺

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem clnbgr3stgrgrlic

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clnbgr3stgrgrlim.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	fvexi 6850	. . . . . . 7 ⊢ 𝑉 ∈ V
3		clnbgr3stgrgrlim.w	. . . . . . . 8 ⊢ 𝑊 = (Vtx‘𝐻)
4	3	fvexi 6850	. . . . . . 7 ⊢ 𝑊 ∈ V
5	2, 4	pm3.2i 470	. . . . . 6 ⊢ (𝑉 ∈ V ∧ 𝑊 ∈ V)
6		breng 8897	. . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝑊 ∈ V) → (𝑉 ≈ 𝑊 ↔ ∃𝑓 𝑓:𝑉–1-1-onto→𝑊))
7	5, 6	mp1i 13	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝑉 ≈ 𝑊 ↔ ∃𝑓 𝑓:𝑉–1-1-onto→𝑊))
8		usgruhgr 29273	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐻 ∈ USGraph → 𝐻 ∈ UHGraph)
9	8	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → 𝐻 ∈ UHGraph)
10	9	3ad2ant1 1134	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐻 ∈ UHGraph)
11	3	clnbgrssvtx 48325	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐻 ClNeighbVtx (𝑓‘𝑥)) ⊆ 𝑊
12	11	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥 ∈ 𝑉) → (𝐻 ClNeighbVtx (𝑓‘𝑥)) ⊆ 𝑊)
13	3	isubgruhgr 48362	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐻 ∈ UHGraph ∧ (𝐻 ClNeighbVtx (𝑓‘𝑥)) ⊆ 𝑊) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ∈ UHGraph)
14	10, 12, 13	syl2an2r 686	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥 ∈ 𝑉) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ∈ UHGraph)
15		f1of 6776	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓:𝑉–1-1-onto→𝑊 → 𝑓:𝑉⟶𝑊)
16	15	3ad2ant2 1135	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ 𝑥 ∈ 𝑉) → 𝑓:𝑉⟶𝑊)
17		simp3 1139	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
18	16, 17	ffvelcdmd 7033	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ 𝑥 ∈ 𝑉) → (𝑓‘𝑥) ∈ 𝑊)
19		oveq2 7370	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = (𝑓‘𝑥) → (𝐻 ClNeighbVtx 𝑦) = (𝐻 ClNeighbVtx (𝑓‘𝑥)))
20	19	oveq2d 7378	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = (𝑓‘𝑥) → (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) = (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))
21	20	breq1d 5096	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝑓‘𝑥) → ((𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) ↔ (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))
22	21	rspcv 3561	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓‘𝑥) ∈ 𝑊 → (∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))
23	18, 22	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ 𝑥 ∈ 𝑉) → (∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))
24	23	3exp 1120	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝑓:𝑉–1-1-onto→𝑊 → (𝑥 ∈ 𝑉 → (∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))))
25	24	com34 91	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝑓:𝑉–1-1-onto→𝑊 → (∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝑥 ∈ 𝑉 → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))))
26	25	3imp1 1349	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥 ∈ 𝑉) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁))
27		gricsym 48415	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ∈ UHGraph → ((𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁) → (StarGr‘𝑁) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))
28	14, 26, 27	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥 ∈ 𝑉) → (StarGr‘𝑁) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))
29	28	anim1ci 617	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥 ∈ 𝑉) ∧ (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁)) → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ (StarGr‘𝑁) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))
30		grictr 48417	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ (StarGr‘𝑁) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))
31	29, 30	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥 ∈ 𝑉) ∧ (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))
32	31	ex 412	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) ∧ 𝑥 ∈ 𝑉) → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))
33	32	ralimdva 3150	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → (∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) → ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))
34	33	3exp 1120	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝑓:𝑉–1-1-onto→𝑊 → (∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) → ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))))
35	34	com24 95	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) → (∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) → (𝑓:𝑉–1-1-onto→𝑊 → ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))))
36	35	imp32 418	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ (∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁))) → (𝑓:𝑉–1-1-onto→𝑊 → ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))
37	36	ancld 550	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ (∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁))) → (𝑓:𝑉–1-1-onto→𝑊 → (𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))))
38	37	eximdv 1919	. . . . . . 7 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ (∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁))) → (∃𝑓 𝑓:𝑉–1-1-onto→𝑊 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))))
39	38	ex 412	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → ((∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → (∃𝑓 𝑓:𝑉–1-1-onto→𝑊 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))))
40	39	com23 86	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (∃𝑓 𝑓:𝑉–1-1-onto→𝑊 → ((∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))))
41	7, 40	sylbid 240	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝑉 ≈ 𝑊 → ((∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))))
42	41	3impia 1118	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊) → ((∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))))
43	42	3impib 1117	. 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥)))))
44	1, 3	dfgrlic2 48502	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))))
45	44	3adant3 1133	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))))
46	45	3ad2ant1 1134	. 2 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))))
47	43, 46	mpbird 257	1 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph ∧ 𝑉 ≈ 𝑊) ∧ ∀𝑥 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑥)) ≃𝑔𝑟 (StarGr‘𝑁) ∧ ∀𝑦 ∈ 𝑊 (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁)) → 𝐺 ≃𝑙𝑔𝑟 𝐻)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ⊆ wss 3890   class class class wbr 5086  ⟶wf 6490  –1-1-onto→wf1o 6493  ‘cfv 6494  (class class class)co 7362   ≈ cen 8885  ℕ₀cn0 12432  Vtxcvtx 29083  UHGraphcuhgr 29143  USGraphcusgr 29236   ClNeighbVtx cclnbgr 48312   ISubGr cisubgr 48354   ≃_𝑔𝑟 cgric 48370  StarGrcstgr 48445   ≃_𝑙𝑔𝑟 cgrlic 48471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-i2m1 11101  ax-1ne0 11102  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-po 5534  df-so 5535  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7937  df-2nd 7938  df-1o 8400  df-er 8638  df-map 8770  df-en 8889  df-dom 8890  df-sdom 8891  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-2 12239  df-vtx 29085  df-iedg 29086  df-uhgr 29145  df-upgr 29169  df-uspgr 29237  df-usgr 29238  df-clnbgr 48313  df-isubgr 48355  df-grim 48372  df-gric 48375  df-grlim 48472  df-grlic 48475

This theorem is referenced by:  gpg5grlic  48588