Theorem grlicsym 49326

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 49322	. . 3 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝑆 → ∃𝑓(𝑓:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))))
4		grlicrcl 49320	. . 3 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
5		vex 3458	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
6		cnvexg 7963	. . . . . . . . . 10 ⊢ (𝑓 ∈ V → ◡𝑓 ∈ V)
7	5, 6	mp1i 14	. . . . . . . . 9 ⊢ (((𝑓:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) ∧ 𝐺 ∈ UHGraph) → ◡𝑓 ∈ V)
8		f1ocnv 6862	. . . . . . . . . . 11 ⊢ (𝑓:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝑆) → ◡𝑓:(Vtx‘𝑆)⟶1-1onto(Vtx‘𝐺))
9	8	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝑓:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) ∧ 𝐺 ∈ UHGraph) → ◡𝑓:(Vtx‘𝑆)⟶1-1onto(Vtx‘𝐺))
10		f1ocnvdm 7319	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆)) → (◡𝑓‘𝑤) ∈ (Vtx‘𝐺))
11	10	3adant3 1150	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (◡𝑓‘𝑤) ∈ (Vtx‘𝐺))
12		oveq2 7454	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝐺 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))
13	12	oveq2d 7462	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) = (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))))
14		fveq2 6911	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝑓‘𝑣) = (𝑓‘(◡𝑓‘𝑤)))
15	14	oveq2d 7462	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝑆 ClNeighbVtx (𝑓‘𝑣)) = (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤))))
16	15	oveq2d 7462	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = (◡𝑓‘𝑤) → (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣))) = (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤)))))
17	13, 16	breq12d 5119	. . . . . . . . . . . . . . . . 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 3576	. . . . . . . . . . . . . . . 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)⟶1-1onto(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 7311	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆)) → (𝑓‘(◡𝑓‘𝑤)) = 𝑤)
21	20	3adant3 1150	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝑓‘(◡𝑓‘𝑤)) = 𝑤)
22	21	oveq2d 7462	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤))) = (𝑆 ClNeighbVtx 𝑤))
23	22	oveq2d 7462	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(◡𝑓‘𝑤)))) = (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)))
24	23	breq2d 5118	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(Vtx‘𝐺)⟶1-1onto(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)⟶1-1onto(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → 𝐺 ∈ UHGraph)
26	1	clnbgrssvtx 49144	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ClNeighbVtx (◡𝑓‘𝑤)) ⊆ (Vtx‘𝐺)
27	1	isubgruhgr 49181	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐺 ClNeighbVtx (◡𝑓‘𝑤)) ⊆ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ∈ UHGraph)
28	25, 26, 27	sylancl 598	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))) ∈ UHGraph)
29		gricsym 49234	. . . . . . . . . . . . . . . . 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)⟶1-1onto(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)⟶1-1onto(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)⟶1-1onto(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)⟶1-1onto(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)⟶1-1onto(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 423	. . . . . . . . . . 11 ⊢ (((𝑓:(Vtx‘𝐺)⟶1-1onto(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 3157	. . . . . . . . . 10 ⊢ (((𝑓:(Vtx‘𝐺)⟶1-1onto(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 521	. . . . . . . . 9 ⊢ (((𝑓:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) ∧ 𝐺 ∈ UHGraph) → (◡𝑓:(Vtx‘𝑆)⟶1-1onto(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))))
38		f1oeq1 6837	. . . . . . . . . 10 ⊢ (𝑔 = ◡𝑓 → (𝑔:(Vtx‘𝑆)⟶1-1onto(Vtx‘𝐺) ↔ ◡𝑓:(Vtx‘𝑆)⟶1-1onto(Vtx‘𝐺)))
39		fveq1 6910	. . . . . . . . . . . . . 14 ⊢ (𝑔 = ◡𝑓 → (𝑔‘𝑤) = (◡𝑓‘𝑤))
40	39	oveq2d 7462	. . . . . . . . . . . . 13 ⊢ (𝑔 = ◡𝑓 → (𝐺 ClNeighbVtx (𝑔‘𝑤)) = (𝐺 ClNeighbVtx (◡𝑓‘𝑤)))
41	40	oveq2d 7462	. . . . . . . . . . . 12 ⊢ (𝑔 = ◡𝑓 → (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔‘𝑤))) = (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))))
42	41	breq2d 5118	. . . . . . . . . . 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 3189	. . . . . . . . . 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 644	. . . . . . . . 9 ⊢ (𝑔 = ◡𝑓 → ((𝑔:(Vtx‘𝑆)⟶1-1onto(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔‘𝑤)))) ↔ (◡𝑓:(Vtx‘𝑆)⟶1-1onto(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (◡𝑓‘𝑤))))))
45	7, 37, 44	spcedv 3556	. . . . . . . 8 ⊢ (((𝑓:(Vtx‘𝐺)⟶1-1onto(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘𝑣)))) ∧ 𝐺 ∈ UHGraph) → ∃𝑔(𝑔:(Vtx‘𝑆)⟶1-1onto(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)⟶1-1onto(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)⟶1-1onto(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔‘𝑤)))))
47	2, 1	dfgrlic2 49321	. . . . . . . . 9 ⊢ ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 ≃𝑙𝑔𝑟 𝐺 ↔ ∃𝑔(𝑔:(Vtx‘𝑆)⟶1-1onto(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔‘𝑤))))))
48	47	ancoms 464	. . . . . . . 8 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝑆 ≃𝑙𝑔𝑟 𝐺 ↔ ∃𝑔(𝑔:(Vtx‘𝑆)⟶1-1onto(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)⟶1-1onto(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)⟶1-1onto(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)⟶1-1onto(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)⟶1-1onto(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)⟶1-1onto(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 1965	. . 3 ⊢ (∃𝑓(𝑓:(Vtx‘𝐺)⟶1-1onto(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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   class class class wbr 5106  ^◡ccnv 5655  ⟶1-1ontowf1o 6560  ‘cfv 6561  (class class class)co 7446  Vtxcvtx 29607  UHGraphcuhgr 29667   ClNeighbVtx cclnbgr 49131   ISubGr cisubgr 49173   ≃_𝑔𝑟 cgric 49189   ≃_𝑙𝑔𝑟 cgrlic 49290

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-1o 8507  df-map 8880  df-vtx 29609  df-iedg 29610  df-uhgr 29669  df-clnbgr 49132  df-isubgr 49174  df-grim 49191  df-gric 49194  df-grlim 49291  df-grlic 49294

This theorem is used by:  grlicsymb  49327  grlicer  49329  usgrexmpl12ngrlic  49352