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Theorem grlicsym 47909
Description: Graph local isomorphism is symmetric for hypergraphs. (Contributed by AV, 9-Jun-2025.)
Assertion
Ref Expression
grlicsym (𝐺 ∈ UHGraph → (𝐺𝑙𝑔𝑟 𝑆𝑆𝑙𝑔𝑟 𝐺))

Proof of Theorem grlicsym
Dummy variables 𝑓 𝑣 𝑔 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2735 . . . 4 (Vtx‘𝑆) = (Vtx‘𝑆)
31, 2grilcbri 47905 . . 3 (𝐺𝑙𝑔𝑟 𝑆 → ∃𝑓(𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣)))))
4 grlicrcl 47903 . . 3 (𝐺𝑙𝑔𝑟 𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
5 vex 3482 . . . . . . . . . 10 𝑓 ∈ V
6 cnvexg 7947 . . . . . . . . . 10 (𝑓 ∈ V → 𝑓 ∈ V)
75, 6mp1i 13 . . . . . . . . 9 (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣)))) ∧ 𝐺 ∈ UHGraph) → 𝑓 ∈ V)
8 f1ocnv 6861 . . . . . . . . . . 11 (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) → 𝑓:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺))
98ad2antrr 726 . . . . . . . . . 10 (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣)))) ∧ 𝐺 ∈ UHGraph) → 𝑓:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺))
10 f1ocnvdm 7305 . . . . . . . . . . . . . . . . 17 ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆)) → (𝑓𝑤) ∈ (Vtx‘𝐺))
11103adant3 1131 . . . . . . . . . . . . . . . 16 ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝑓𝑤) ∈ (Vtx‘𝐺))
12 oveq2 7439 . . . . . . . . . . . . . . . . . . 19 (𝑣 = (𝑓𝑤) → (𝐺 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx (𝑓𝑤)))
1312oveq2d 7447 . . . . . . . . . . . . . . . . . 18 (𝑣 = (𝑓𝑤) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) = (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤))))
14 fveq2 6907 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = (𝑓𝑤) → (𝑓𝑣) = (𝑓‘(𝑓𝑤)))
1514oveq2d 7447 . . . . . . . . . . . . . . . . . . 19 (𝑣 = (𝑓𝑤) → (𝑆 ClNeighbVtx (𝑓𝑣)) = (𝑆 ClNeighbVtx (𝑓‘(𝑓𝑤))))
1615oveq2d 7447 . . . . . . . . . . . . . . . . . 18 (𝑣 = (𝑓𝑤) → (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣))) = (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(𝑓𝑤)))))
1713, 16breq12d 5161 . . . . . . . . . . . . . . . . 17 (𝑣 = (𝑓𝑤) → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣))) ↔ (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(𝑓𝑤))))))
1817rspcv 3618 . . . . . . . . . . . . . . . 16 ((𝑓𝑤) ∈ (Vtx‘𝐺) → (∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣))) → (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(𝑓𝑤))))))
1911, 18syl 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 (𝑓𝑣))) → (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(𝑓𝑤))))))
20 f1ocnvfv2 7297 . . . . . . . . . . . . . . . . . . . 20 ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆)) → (𝑓‘(𝑓𝑤)) = 𝑤)
21203adant3 1131 . . . . . . . . . . . . . . . . . . 19 ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝑓‘(𝑓𝑤)) = 𝑤)
2221oveq2d 7447 . . . . . . . . . . . . . . . . . 18 ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝑆 ClNeighbVtx (𝑓‘(𝑓𝑤))) = (𝑆 ClNeighbVtx 𝑤))
2322oveq2d 7447 . . . . . . . . . . . . . . . . 17 ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(𝑓𝑤)))) = (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)))
2423breq2d 5160 . . . . . . . . . . . . . . . 16 ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → ((𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(𝑓𝑤)))) ↔ (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤))))
25 simp3 1137 . . . . . . . . . . . . . . . . . 18 ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → 𝐺 ∈ UHGraph)
261clnbgrssvtx 47756 . . . . . . . . . . . . . . . . . 18 (𝐺 ClNeighbVtx (𝑓𝑤)) ⊆ (Vtx‘𝐺)
271isubgruhgr 47792 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ UHGraph ∧ (𝐺 ClNeighbVtx (𝑓𝑤)) ⊆ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤))) ∈ UHGraph)
2825, 26, 27sylancl 586 . . . . . . . . . . . . . . . . 17 ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤))) ∈ UHGraph)
29 gricsym 47828 . . . . . . . . . . . . . . . . 17 ((𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤))) ∈ UHGraph → ((𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) → (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤)))))
3028, 29syl 17 . . . . . . . . . . . . . . . 16 ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → ((𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) → (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤)))))
3124, 30sylbid 240 . . . . . . . . . . . . . . 15 ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ 𝑤 ∈ (Vtx‘𝑆) ∧ 𝐺 ∈ UHGraph) → ((𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤))) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓‘(𝑓𝑤)))) → (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤)))))
3219, 31syld 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 (𝑓𝑣))) → (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤)))))
33323exp 1118 . . . . . . . . . . . . 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 (𝑓𝑣))) → (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤)))))))
3433com24 95 . . . . . . . . . . . 12 (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) → (∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣))) → (𝐺 ∈ UHGraph → (𝑤 ∈ (Vtx‘𝑆) → (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤)))))))
3534imp31 417 . . . . . . . . . . 11 (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣)))) ∧ 𝐺 ∈ UHGraph) → (𝑤 ∈ (Vtx‘𝑆) → (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤)))))
3635ralrimiv 3143 . . . . . . . . . 10 (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣)))) ∧ 𝐺 ∈ UHGraph) → ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤))))
379, 36jca 511 . . . . . . . . 9 (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣)))) ∧ 𝐺 ∈ UHGraph) → (𝑓:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤)))))
38 f1oeq1 6837 . . . . . . . . . 10 (𝑔 = 𝑓 → (𝑔:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ↔ 𝑓:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺)))
39 fveq1 6906 . . . . . . . . . . . . . 14 (𝑔 = 𝑓 → (𝑔𝑤) = (𝑓𝑤))
4039oveq2d 7447 . . . . . . . . . . . . 13 (𝑔 = 𝑓 → (𝐺 ClNeighbVtx (𝑔𝑤)) = (𝐺 ClNeighbVtx (𝑓𝑤)))
4140oveq2d 7447 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔𝑤))) = (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤))))
4241breq2d 5160 . . . . . . . . . . 11 (𝑔 = 𝑓 → ((𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔𝑤))) ↔ (𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤)))))
4342ralbidv 3176 . . . . . . . . . 10 (𝑔 = 𝑓 → (∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔𝑤))) ↔ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤)))))
4438, 43anbi12d 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 (𝑔𝑤)))) ↔ (𝑓:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓𝑤))))))
457, 37, 44spcedv 3598 . . . . . . . 8 (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣)))) ∧ 𝐺 ∈ UHGraph) → ∃𝑔(𝑔:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔𝑤)))))
46453adant3 1131 . . . . . . 7 (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣)))) ∧ 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 (𝑔𝑤)))))
472, 1dfgrlic2 47904 . . . . . . . . 9 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆𝑙𝑔𝑟 𝐺 ↔ ∃𝑔(𝑔:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔𝑤))))))
4847ancoms 458 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝑆𝑙𝑔𝑟 𝐺 ↔ ∃𝑔(𝑔:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔𝑤))))))
49483ad2ant3 1134 . . . . . . 7 (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣)))) ∧ 𝐺 ∈ UHGraph ∧ (𝐺 ∈ V ∧ 𝑆 ∈ V)) → (𝑆𝑙𝑔𝑟 𝐺 ↔ ∃𝑔(𝑔:(Vtx‘𝑆)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑤 ∈ (Vtx‘𝑆)(𝑆 ISubGr (𝑆 ClNeighbVtx 𝑤)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑔𝑤))))))
5046, 49mpbird 257 . . . . . 6 (((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣)))) ∧ 𝐺 ∈ UHGraph ∧ (𝐺 ∈ V ∧ 𝑆 ∈ V)) → 𝑆𝑙𝑔𝑟 𝐺)
51503exp 1118 . . . . 5 ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣)))) → (𝐺 ∈ UHGraph → ((𝐺 ∈ V ∧ 𝑆 ∈ V) → 𝑆𝑙𝑔𝑟 𝐺)))
5251com23 86 . . . 4 ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣)))) → ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝐺 ∈ UHGraph → 𝑆𝑙𝑔𝑟 𝐺)))
5352exlimiv 1928 . . 3 (∃𝑓(𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝑆) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝑆 ISubGr (𝑆 ClNeighbVtx (𝑓𝑣)))) → ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝐺 ∈ UHGraph → 𝑆𝑙𝑔𝑟 𝐺)))
543, 4, 53sylc 65 . 2 (𝐺𝑙𝑔𝑟 𝑆 → (𝐺 ∈ UHGraph → 𝑆𝑙𝑔𝑟 𝐺))
5554com12 32 1 (𝐺 ∈ UHGraph → (𝐺𝑙𝑔𝑟 𝑆𝑆𝑙𝑔𝑟 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wral 3059  Vcvv 3478  wss 3963   class class class wbr 5148  ccnv 5688  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  Vtxcvtx 29028  UHGraphcuhgr 29088   ClNeighbVtx cclnbgr 47743   ISubGr cisubgr 47784  𝑔𝑟 cgric 47800  𝑙𝑔𝑟 cgrlic 47880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-1o 8505  df-map 8867  df-vtx 29030  df-iedg 29031  df-uhgr 29090  df-clnbgr 47744  df-isubgr 47785  df-grim 47802  df-gric 47805  df-grlim 47881  df-grlic 47884
This theorem is referenced by:  grlicsymb  47910  grlicer  47912  usgrexmpl12ngrlic  47934
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