Theorem grlicref 49204

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 6901	. . . 4 ⊢ (𝐺 ∈ UHGraph → (Vtx‘𝐺) ∈ V)
2	1	resiexd 7223	. . 3 ⊢ (𝐺 ∈ UHGraph → ( I ↾ (Vtx‘𝐺)) ∈ V)
3		eqid 2760	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	clnbgrssvtx 49023	. . . . . . . 8 ⊢ (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺)
5	4	a1i 11	. . . . . . 7 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺))
6	3	isubgruhgr 49060	. . . . . . 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 49112	. . . . . 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 3154	. . . 4 ⊢ (𝐺 ∈ UHGraph → ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))
11		f1oi 6864	. . . 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 6813	. . . 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 6885	. . . . . . . . 9 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝑓‘𝑣) = (( I ↾ (Vtx‘𝐺))‘𝑣))
15	14	oveq2d 7437	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx (𝑓‘𝑣)) = (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣)))
16	15	oveq2d 7437	. . . . . . 7 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓‘𝑣))) = (𝐺 ISubGr (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣))))
17	16	breq2d 5114	. . . . . 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 7179	. . . . . . . . 9 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (( I ↾ (Vtx‘𝐺))‘𝑣) = 𝑣)
19	18	oveq2d 7437	. . . . . . . 8 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣)) = (𝐺 ClNeighbVtx 𝑣))
20	19	oveq2d 7437	. . . . . . 7 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (𝐺 ISubGr (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣))) = (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))
21	20	breq2d 5114	. . . . . 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 3183	. . . 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 3552	. 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 49200	. . 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 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   class class class wbr 5102   I cid 5544   ↾ cres 5652  –1-1-onto→wf1o 6538  ‘cfv 6539  (class class class)co 7421  Vtxcvtx 29580  UHGraphcuhgr 29640   ClNeighbVtx cclnbgr 49010   ISubGr cisubgr 49052   ≃_𝑔𝑟 cgric 49068   ≃_𝑙𝑔𝑟 cgrlic 49169

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-1o 8480  df-map 8853  df-vtx 29582  df-iedg 29583  df-uhgr 29642  df-clnbgr 49011  df-isubgr 49053  df-grim 49070  df-gric 49073  df-grlim 49170  df-grlic 49173

This theorem is used by:  grlicer  49208