Theorem grlicref 48818

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 6903	. . . 4 ⊢ (𝐺 ∈ UHGraph → (Vtx‘𝐺) ∈ V)
2	1	resiexd 7221	. . 3 ⊢ (𝐺 ∈ UHGraph → ( I ↾ (Vtx‘𝐺)) ∈ V)
3		eqid 2766	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	clnbgrssvtx 48637	. . . . . . . 8 ⊢ (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺)
5	4	a1i 11	. . . . . . 7 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺))
6	3	isubgruhgr 48674	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ∈ UHGraph)
7	5, 6	sylan2 605	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ∈ UHGraph)
8		gricref 48726	. . . . . 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 3160	. . . 4 ⊢ (𝐺 ∈ UHGraph → ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))
11		f1oi 6866	. . . 4 ⊢ ( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺)
12	10, 11	jctil 529	. . 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 6815	. . . 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 6887	. . . . . . . . 9 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝑓‘𝑣) = (( I ↾ (Vtx‘𝐺))‘𝑣))
15	14	oveq2d 7439	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx (𝑓‘𝑣)) = (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣)))
16	15	oveq2d 7439	. . . . . . 7 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓‘𝑣))) = (𝐺 ISubGr (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣))))
17	16	breq2d 5126	. . . . . 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 7178	. . . . . . . . 9 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (( I ↾ (Vtx‘𝐺))‘𝑣) = 𝑣)
19	18	oveq2d 7439	. . . . . . . 8 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣)) = (𝐺 ClNeighbVtx 𝑣))
20	19	oveq2d 7439	. . . . . . 7 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (𝐺 ISubGr (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣))) = (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))
21	20	breq2d 5126	. . . . . 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 519	. . . . 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 3189	. . . 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 644	. . 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 48814	. . 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 577	. 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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∀wral 3082  Vcvv 3458   ⊆ wss 3908   class class class wbr 5114   I cid 5560   ↾ cres 5668  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7423  Vtxcvtx 29383  UHGraphcuhgr 29443   ClNeighbVtx cclnbgr 48624   ISubGr cisubgr 48666   ≃_𝑔𝑟 cgric 48682   ≃_𝑙𝑔𝑟 cgrlic 48783

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-1o 8462  df-map 8835  df-vtx 29385  df-iedg 29386  df-uhgr 29445  df-clnbgr 48625  df-isubgr 48667  df-grim 48684  df-gric 48687  df-grlim 48784  df-grlic 48787

This theorem is used by:  grlicer  48822