Theorem clnbgr3stgrgrlic 48923

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 6896	. . . . . . 7 ⊢ 𝑉 ∈ V
3		clnbgr3stgrgrlim.w	. . . . . . . 8 ⊢ 𝑊 = (Vtx‘𝐻)
4	3	fvexi 6896	. . . . . . 7 ⊢ 𝑊 ∈ V
5	2, 4	pm3.2i 476	. . . . . 6 ⊢ (𝑉 ∈ V ∧ 𝑊 ∈ V)
6		breng 8964	. . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝑊 ∈ V) → (𝑉 ≈ 𝑊 ↔ ∃𝑓 𝑓:𝑉–1-1-onto→𝑊))
7	5, 6	mp1i 14	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → (𝑉 ≈ 𝑊 ↔ ∃𝑓 𝑓:𝑉–1-1-onto→𝑊))
8		usgruhgr 29632	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐻 ∈ USGraph → 𝐻 ∈ UHGraph)
9	8	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) → 𝐻 ∈ UHGraph)
10	9	3ad2ant1 1151	. . . . . . . . . . . . . . . . . 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 48734	. . . . . . . . . . . . . . . . . . 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 48771	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐻 ∈ UHGraph ∧ (𝐻 ClNeighbVtx (𝑓‘𝑥)) ⊆ 𝑊) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ∈ UHGraph)
14	10, 12, 13	syl2an2r 698	. . . . . . . . . . . . . . . . 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 6821	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓:𝑉–1-1-onto→𝑊 → 𝑓:𝑉⟶𝑊)
16	15	3ad2ant2 1152	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ 𝑥 ∈ 𝑉) → 𝑓:𝑉⟶𝑊)
17		simp3 1156	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
18	16, 17	ffvelcdmd 7081	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ USGraph ∧ 𝐻 ∈ USGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊 ∧ 𝑥 ∈ 𝑉) → (𝑓‘𝑥) ∈ 𝑊)
19		oveq2 7424	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = (𝑓‘𝑥) → (𝐻 ClNeighbVtx 𝑦) = (𝐻 ClNeighbVtx (𝑓‘𝑥)))
20	19	oveq2d 7432	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = (𝑓‘𝑥) → (𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) = (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))))
21	20	breq1d 5117	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝑓‘𝑥) → ((𝐻 ISubGr (𝐻 ClNeighbVtx 𝑦)) ≃𝑔𝑟 (StarGr‘𝑁) ↔ (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑥))) ≃𝑔𝑟 (StarGr‘𝑁)))
22	21	rspcv 3575	. . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . . . . . 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 92	. . . . . . . . . . . . . . . . . 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 1366	. . . . . . . . . . . . . . . . 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 48824	. . . . . . . . . . . . . . . . 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 66	. . . . . . . . . . . . . . . 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 628	. . . . . . . . . . . . . . 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 48826	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 418	. . . . . . . . . . . . 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 3176	. . . . . . . . . . . 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 1137	. . . . . . . . . . 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 96	. . . . . . . . . 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 424	. . . . . . . . 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 560	. . . . . . . 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 1950	. . . . . . 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 418	. . . . . 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 87	. . . . 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 243	. . . 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 1135	. . 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 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)) → ∃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 48911	. . . 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 1150	. . 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 1151	. 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 260	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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   ⊆ wss 3902   class class class wbr 5107  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7416   ≈ cen 8952  ℕ₀cn0 12531  Vtxcvtx 29439  UHGraphcuhgr 29499  USGraphcusgr 29595   ClNeighbVtx cclnbgr 48721   ISubGr cisubgr 48763   ≃_𝑔𝑟 cgric 48779  StarGrcstgr 48854   ≃_𝑙𝑔𝑟 cgrlic 48880

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-i2m1 11195  ax-1ne0 11196  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-po 5567  df-so 5568  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-1o 8458  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-2 12330  df-vtx 29441  df-iedg 29442  df-uhgr 29501  df-upgr 29525  df-uspgr 29596  df-usgr 29597  df-clnbgr 48722  df-isubgr 48764  df-grim 48781  df-gric 48784  df-grlim 48881  df-grlic 48884

This theorem is used by:  gpg5grlic  48997