Theorem grlicsym 48255

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 2736	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2736	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
3	1, 2	grilcbri 48251	. . 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 48249	. . 3 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
5		vex 3444	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
6		cnvexg 7866	. . . . . . . . . 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 6786	. . . . . . . . . . 11 ⊢ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) → ◡𝑓:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺))
9	8	ad2antrr 726	. . . . . . . . . 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 7231	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆)) → (◡𝑓‘𝑤) ∈ (Vtx‘𝐺))
11	10	3adant3 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (◡𝑓‘𝑤) ∈ (Vtx‘𝐺))
12		oveq2 7366	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝐺 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))
13	12	oveq2d 7374	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) = (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))))
14		fveq2 6834	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝑓‘𝑣) = (𝑓‘(◡𝑓‘𝑤)))
15	14	oveq2d 7374	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝑆 ClNeighbVtx (𝑓‘𝑣)) = (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤))))
16	15	oveq2d 7374	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣))) = (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤)))))
17	13, 16	breq12d 5111	. . . . . . . . . . . . . . . . 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 3572	. . . . . . . . . . . . . . . 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 7223	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆)) → (𝑓‘(◡𝑓‘𝑤)) = 𝑤)
21	20	3adant3 1132	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝑓‘(◡𝑓‘𝑤)) = 𝑤)
22	21	oveq2d 7374	. . . . . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . . . . . 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 5110	. . . . . . . . . . . . . . . 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 1138	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → 𝐺 ∈ UHGraph)
26	1	clnbgrssvtx 48073	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ClNeighbVtx (◡𝑓‘𝑤)) ⊆ (Vtx‘𝐺)
27	1	isubgruhgr 48110	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐺 ClNeighbVtx (◡𝑓‘𝑤)) ⊆ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ∈ UHGraph)
28	25, 26, 27	sylancl 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ∈ UHGraph)
29		gricsym 48163	. . . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . 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 417	. . . . . . . . . . 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 3127	. . . . . . . . . 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 511	. . . . . . . . 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 6762	. . . . . . . . . 10 ⊢ (𝑔 = ◡𝑓 → (𝑔:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ↔ ◡𝑓:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺)))
39		fveq1 6833	. . . . . . . . . . . . . 14 ⊢ (𝑔 = ◡𝑓 → (𝑔‘𝑤) = (◡𝑓‘𝑤))
40	39	oveq2d 7374	. . . . . . . . . . . . 13 ⊢ (𝑔 = ◡𝑓 → (𝐺 ClNeighbVtx (𝑔‘𝑤)) = (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))
41	40	oveq2d 7374	. . . . . . . . . . . 12 ⊢ (𝑔 = ◡𝑓 → (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔‘𝑤))) = (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))))
42	41	breq2d 5110	. . . . . . . . . . 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 3159	. . . . . . . . . 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 632	. . . . . . . . 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 3552	. . . . . . . 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 1132	. . . . . . 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 48250	. . . . . . . . 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 458	. . . . . . . 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 1135	. . . . . . 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 257	. . . . . 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 1119	. . . . 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 1931	. . 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3051  Vcvv 3440   ⊆ wss 3901   class class class wbr 5098  ^◡ccnv 5623  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358  Vtxcvtx 29069  UHGraphcuhgr 29129   ClNeighbVtx cclnbgr 48060   ISubGr cisubgr 48102   ≃_𝑔𝑟 cgric 48118   ≃_𝑙𝑔𝑟 cgrlic 48219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-1o 8397  df-map 8765  df-vtx 29071  df-iedg 29072  df-uhgr 29131  df-clnbgr 48061  df-isubgr 48103  df-grim 48120  df-gric 48123  df-grlim 48220  df-grlic 48223

This theorem is referenced by:  grlicsymb  48256  grlicer  48258  usgrexmpl12ngrlic  48281