Theorem grlicsym 48755

Description: Graph local isomorphism is symmetric for hypergraphs. (Contributed by AV, 9-Jun-2025.)

Assertion

Ref	Expression
grlicsym	⊢ (𝐺 ∈ UHGraph → (𝐺 ≃𝑙𝑔𝑟 𝑆 → 𝑆 ≃𝑙𝑔𝑟 𝐺))

Proof of Theorem grlicsym

Dummy variables 𝑓 𝑣 𝑔 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2763	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
3	1, 2	grilcbri 48751	. . 3 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝑆 → ∃𝑓(𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))))
4		grlicrcl 48749	. . 3 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
5		vex 3459	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
6		cnvexg 7922	. . . . . . . . . 10 ⊢ (𝑓 ∈ V → ◡𝑓 ∈ V)
7	5, 6	mp1i 14	. . . . . . . . 9 ⊢ (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) ∧ 𝐺 ∈ UHGraph) → ◡𝑓 ∈ V)
8		f1ocnv 6835	. . . . . . . . . . 11 ⊢ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) → ◡𝑓:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺))
9	8	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) ∧ 𝐺 ∈ UHGraph) → ◡𝑓:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺))
10		f1ocnvdm 7285	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆)) → (◡𝑓‘𝑤) ∈ (Vtx‘𝐺))
11	10	3adant3 1150	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (◡𝑓‘𝑤) ∈ (Vtx‘𝐺))
12		oveq2 7420	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝐺 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))
13	12	oveq2d 7428	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) = (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))))
14		fveq2 6883	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝑓‘𝑣) = (𝑓‘(◡𝑓‘𝑤)))
15	14	oveq2d 7428	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝑆 ClNeighbVtx (𝑓‘𝑣)) = (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤))))
16	15	oveq2d 7428	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣))) = (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤)))))
17	13, 16	breq12d 5123	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (◡𝑓‘𝑤) → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣))) ↔ (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤))))))
18	17	rspcv 3578	. . . . . . . . . . . . . . . 16 ⊢ ((◡𝑓‘𝑤) ∈ (Vtx‘𝐺) → (∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣))) → (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤))))))
19	11, 18	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣))) → (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤))))))
20		f1ocnvfv2 7277	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆)) → (𝑓‘(◡𝑓‘𝑤)) = 𝑤)
21	20	3adant3 1150	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝑓‘(◡𝑓‘𝑤)) = 𝑤)
22	21	oveq2d 7428	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤))) = (𝑆 ClNeighbVtx 𝑤))
23	22	oveq2d 7428	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤)))) = (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)))
24	23	breq2d 5122	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → ((𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤)))) ↔ (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤))))
25		simp3 1156	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → 𝐺 ∈ UHGraph)
26	1	clnbgrssvtx 48573	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ClNeighbVtx (◡𝑓‘𝑤)) ⊆ (Vtx‘𝐺)
27	1	isubgruhgr 48610	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐺 ClNeighbVtx (◡𝑓‘𝑤)) ⊆ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ∈ UHGraph)
28	25, 26, 27	sylancl 597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ∈ UHGraph)
29		gricsym 48663	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ∈ UHGraph → ((𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) → (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))))
30	28, 29	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → ((𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) → (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))))
31	24, 30	sylbid 243	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → ((𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤)))) → (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))))
32	19, 31	syld 48	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣))) → (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))))
33	32	3exp 1137	. . . . . . . . . . . . 13 ⊢ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) → (𝑤 ∈ (Vtx‘𝑆) → (𝐺 ∈ UHGraph → (∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣))) → (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))))))
34	33	com24 96	. . . . . . . . . . . 12 ⊢ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) → (∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣))) → (𝐺 ∈ UHGraph → (𝑤 ∈ (Vtx‘𝑆) → (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))))))
35	34	imp31 422	. . . . . . . . . . 11 ⊢ (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) ∧ 𝐺 ∈ UHGraph) → (𝑤 ∈ (Vtx‘𝑆) → (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))))
36	35	ralrimiv 3156	. . . . . . . . . 10 ⊢ (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) ∧ 𝐺 ∈ UHGraph) → ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))))
37	9, 36	jca 520	. . . . . . . . 9 ⊢ (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) ∧ 𝐺 ∈ UHGraph) → (◡𝑓:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))))
38		f1oeq1 6810	. . . . . . . . . 10 ⊢ (𝑔 = ◡𝑓 → (𝑔:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ↔ ◡𝑓:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺)))
39		fveq1 6882	. . . . . . . . . . . . . 14 ⊢ (𝑔 = ◡𝑓 → (𝑔‘𝑤) = (◡𝑓‘𝑤))
40	39	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (𝑔 = ◡𝑓 → (𝐺 ClNeighbVtx (𝑔‘𝑤)) = (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))
41	40	oveq2d 7428	. . . . . . . . . . . 12 ⊢ (𝑔 = ◡𝑓 → (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔‘𝑤))) = (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))))
42	41	breq2d 5122	. . . . . . . . . . 11 ⊢ (𝑔 = ◡𝑓 → ((𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔‘𝑤))) ↔ (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))))
43	42	ralbidv 3188	. . . . . . . . . 10 ⊢ (𝑔 = ◡𝑓 → (∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔‘𝑤))) ↔ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))))
44	38, 43	anbi12d 643	. . . . . . . . 9 ⊢ (𝑔 = ◡𝑓 → ((𝑔:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔‘𝑤)))) ↔ (◡𝑓:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))))))
45	7, 37, 44	spcedv 3558	. . . . . . . 8 ⊢ (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) ∧ 𝐺 ∈ UHGraph) → ∃𝑔(𝑔:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔‘𝑤)))))
46	45	3adant3 1150	. . . . . . 7 ⊢ (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) ∧ 𝐺 ∈ UHGraph ∧ (𝐺 ∈ V ∧ 𝑆 ∈ V)) → ∃𝑔(𝑔:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔‘𝑤)))))
47	2, 1	dfgrlic2 48750	. . . . . . . . 9 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 ≃𝑙𝑔𝑟 𝐺 ↔ ∃𝑔(𝑔:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔‘𝑤))))))
48	47	ancoms 463	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝑆 ≃𝑙𝑔𝑟 𝐺 ↔ ∃𝑔(𝑔:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔‘𝑤))))))
49	48	3ad2ant3 1153	. . . . . . 7 ⊢ (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) ∧ 𝐺 ∈ UHGraph ∧ (𝐺 ∈ V ∧ 𝑆 ∈ V)) → (𝑆 ≃𝑙𝑔𝑟 𝐺 ↔ ∃𝑔(𝑔:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔‘𝑤))))))
50	46, 49	mpbird 260	. . . . . 6 ⊢ (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) ∧ 𝐺 ∈ UHGraph ∧ (𝐺 ∈ V ∧ 𝑆 ∈ V)) → 𝑆 ≃𝑙𝑔𝑟 𝐺)
51	50	3exp 1137	. . . . 5 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) → (𝐺 ∈ UHGraph → ((𝐺 ∈ V ∧ 𝑆 ∈ V) → 𝑆 ≃𝑙𝑔𝑟 𝐺)))
52	51	com23 87	. . . 4 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) → ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝐺 ∈ UHGraph → 𝑆 ≃𝑙𝑔𝑟 𝐺)))
53	52	exlimiv 1960	. . 3 ⊢ (∃𝑓(𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) → ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝐺 ∈ UHGraph → 𝑆 ≃𝑙𝑔𝑟 𝐺)))
54	3, 4, 53	sylc 66	. 2 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝑆 → (𝐺 ∈ UHGraph → 𝑆 ≃𝑙𝑔𝑟 𝐺))
55	54	com12 33	1 ⊢ (𝐺 ∈ UHGraph → (𝐺 ≃𝑙𝑔𝑟 𝑆 → 𝑆 ≃𝑙𝑔𝑟 𝐺))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ⊆ wss 3906   class class class wbr 5110  ^◡ccnv 5662  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7412  Vtxcvtx 29324  UHGraphcuhgr 29384   ClNeighbVtx cclnbgr 48560   ISubGr cisubgr 48602   ≃_𝑔𝑟 cgric 48618   ≃_𝑙𝑔𝑟 cgrlic 48719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-1o 8454  df-map 8827  df-vtx 29326  df-iedg 29327  df-uhgr 29386  df-clnbgr 48561  df-isubgr 48603  df-grim 48620  df-gric 48623  df-grlim 48720  df-grlic 48723

This theorem is referenced by:  grlicsymb  48756  grlicer  48758  usgrexmpl12ngrlic  48781