Theorem grlicref 48633

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

Assertion

Ref	Expression
grlicref	⊢ (𝐺 ∈ UHGraph → 𝐺 ≃𝑙𝑔𝑟 𝐺)

Proof of Theorem grlicref

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

Step	Hyp	Ref	Expression
1		fvexd 6886	. . . 4 ⊢ (𝐺 ∈ UHGraph → (Vtx‘𝐺) ∈ V)
2	1	resiexd 7204	. . 3 ⊢ (𝐺 ∈ UHGraph → ( I ↾ (Vtx‘𝐺)) ∈ V)
3		eqid 2765	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	clnbgrssvtx 48452	. . . . . . . 8 ⊢ (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺)
5	4	a1i 11	. . . . . . 7 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺))
6	3	isubgruhgr 48489	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ∈ UHGraph)
7	5, 6	sylan2 604	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ∈ UHGraph)
8		gricref 48541	. . . . . 6 ⊢ ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ∈ UHGraph → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))
9	7, 8	syl 18	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))
10	9	ralrimiva 3157	. . . 4 ⊢ (𝐺 ∈ UHGraph → ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))
11		f1oi 6849	. . . 4 ⊢ ( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺)
12	10, 11	jctil 528	. . 3 ⊢ (𝐺 ∈ UHGraph → (( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣))))
13		f1oeq1 6798	. . . 4 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺) ↔ ( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺)))
14		fveq1 6870	. . . . . . . . 9 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝑓‘𝑣) = (( I ↾ (Vtx‘𝐺))‘𝑣))
15	14	oveq2d 7416	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx (𝑓‘𝑣)) = (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣)))
16	15	oveq2d 7416	. . . . . . 7 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓‘𝑣))) = (𝐺 ISubGr (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣))))
17	16	breq2d 5116	. . . . . 6 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓‘𝑣))) ↔ (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣)))))
18		fvresi 7161	. . . . . . . . 9 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (( I ↾ (Vtx‘𝐺))‘𝑣) = 𝑣)
19	18	oveq2d 7416	. . . . . . . 8 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣)) = (𝐺 ClNeighbVtx 𝑣))
20	19	oveq2d 7416	. . . . . . 7 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (𝐺 ISubGr (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣))) = (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))
21	20	breq2d 5116	. . . . . 6 ⊢ (𝑣 ∈ (Vtx‘𝐺) → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣))) ↔ (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣))))
22	17, 21	sylan9bb 518	. . . . 5 ⊢ ((𝑓 = ( I ↾ (Vtx‘𝐺)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓‘𝑣))) ↔ (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣))))
23	22	ralbidva 3186	. . . 4 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓‘𝑣))) ↔ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣))))
24	13, 23	anbi12d 643	. . 3 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓‘𝑣)))) ↔ (( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))))
25	2, 12, 24	spcedv 3560	. 2 ⊢ (𝐺 ∈ UHGraph → ∃𝑓(𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓‘𝑣)))))
26	3, 3	dfgrlic2 48629	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐺 ∈ UHGraph) → (𝐺 ≃𝑙𝑔𝑟 𝐺 ↔ ∃𝑓(𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓‘𝑣))))))
27	26	anidms 576	. 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 ≃𝑙𝑔𝑟 𝐺 ↔ ∃𝑓(𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓‘𝑣))))))
28	25, 27	mpbird 260	1 ⊢ (𝐺 ∈ UHGraph → 𝐺 ≃𝑙𝑔𝑟 𝐺)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∀wral 3079  Vcvv 3457   ⊆ wss 3907   class class class wbr 5104   I cid 5545   ↾ cres 5653  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400  Vtxcvtx 29251  UHGraphcuhgr 29311   ClNeighbVtx cclnbgr 48439   ISubGr cisubgr 48481   ≃_𝑔𝑟 cgric 48497   ≃_𝑙𝑔𝑟 cgrlic 48598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-1o 8441  df-map 8814  df-vtx 29253  df-iedg 29254  df-uhgr 29313  df-clnbgr 48440  df-isubgr 48482  df-grim 48499  df-gric 48502  df-grlim 48599  df-grlic 48602

This theorem is referenced by:  grlicer  48637