Theorem grlicsym 48599

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 2761	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2761	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
3	1, 2	grilcbri 48595	. . 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 48593	. . 3 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
5		vex 3457	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
6		cnvexg 7901	. . . . . . . . . 10 ⊢ (𝑓 ∈ V → ◡𝑓 ∈ V)
7	5, 6	mp1i 13	. . . . . . . . 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 6815	. . . . . . . . . . 11 ⊢ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) → ◡𝑓:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺))
9	8	ad2antrr 736	. . . . . . . . . 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 7265	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆)) → (◡𝑓‘𝑤) ∈ (Vtx‘𝐺))
11	10	3adant3 1144	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (◡𝑓‘𝑤) ∈ (Vtx‘𝐺))
12		oveq2 7400	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝐺 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))
13	12	oveq2d 7408	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) = (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))))
14		fveq2 6863	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝑓‘𝑣) = (𝑓‘(◡𝑓‘𝑤)))
15	14	oveq2d 7408	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝑆 ClNeighbVtx (𝑓‘𝑣)) = (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤))))
16	15	oveq2d 7408	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣))) = (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤)))))
17	13, 16	breq12d 5112	. . . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . 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 7257	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆)) → (𝑓‘(◡𝑓‘𝑤)) = 𝑤)
21	20	3adant3 1144	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝑓‘(◡𝑓‘𝑤)) = 𝑤)
22	21	oveq2d 7408	. . . . . . . . . . . . . . . . . 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 7408	. . . . . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . . . . 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 1150	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → 𝐺 ∈ UHGraph)
26	1	clnbgrssvtx 48417	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ClNeighbVtx (◡𝑓‘𝑤)) ⊆ (Vtx‘𝐺)
27	1	isubgruhgr 48454	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐺 ClNeighbVtx (◡𝑓‘𝑤)) ⊆ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ∈ UHGraph)
28	25, 26, 27	sylancl 595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ∈ UHGraph)
29		gricsym 48507	. . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . 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 47	. . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . 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 95	. . . . . . . . . . . 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 421	. . . . . . . . . . 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 3152	. . . . . . . . . 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 519	. . . . . . . . 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 6790	. . . . . . . . . 10 ⊢ (𝑔 = ◡𝑓 → (𝑔:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ↔ ◡𝑓:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺)))
39		fveq1 6862	. . . . . . . . . . . . . 14 ⊢ (𝑔 = ◡𝑓 → (𝑔‘𝑤) = (◡𝑓‘𝑤))
40	39	oveq2d 7408	. . . . . . . . . . . . 13 ⊢ (𝑔 = ◡𝑓 → (𝐺 ClNeighbVtx (𝑔‘𝑤)) = (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))
41	40	oveq2d 7408	. . . . . . . . . . . 12 ⊢ (𝑔 = ◡𝑓 → (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔‘𝑤))) = (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))))
42	41	breq2d 5111	. . . . . . . . . . 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 3184	. . . . . . . . . 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 641	. . . . . . . . 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 3557	. . . . . . . 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 1144	. . . . . . 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 48594	. . . . . . . . 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 462	. . . . . . . 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 1147	. . . . . . 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 259	. . . . . 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 1131	. . . . 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 86	. . . 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 1949	. . 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 65	. 2 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝑆 → (𝐺 ∈ UHGraph → 𝑆 ≃𝑙𝑔𝑟 𝐺))
55	54	com12 32	1 ⊢ (𝐺 ∈ UHGraph → (𝐺 ≃𝑙𝑔𝑟 𝑆 → 𝑆 ≃𝑙𝑔𝑟 𝐺))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ⊆ wss 3904   class class class wbr 5099  ^◡ccnv 5644  –1-1-onto→wf1o 6516  ‘cfv 6517  (class class class)co 7392  Vtxcvtx 29143  UHGraphcuhgr 29203   ClNeighbVtx cclnbgr 48404   ISubGr cisubgr 48446   ≃_𝑔𝑟 cgric 48462   ≃_𝑙𝑔𝑟 cgrlic 48563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-1o 8432  df-map 8805  df-vtx 29145  df-iedg 29146  df-uhgr 29205  df-clnbgr 48405  df-isubgr 48447  df-grim 48464  df-gric 48467  df-grlim 48564  df-grlic 48567

This theorem is referenced by:  grlicsymb  48600  grlicer  48602  usgrexmpl12ngrlic  48625