Theorem grlicref 48594

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 6876	. . . 4 ⊢ (𝐺 ∈ UHGraph → (Vtx‘𝐺) ∈ V)
2	1	resiexd 7194	. . 3 ⊢ (𝐺 ∈ UHGraph → ( I ↾ (Vtx‘𝐺)) ∈ V)
3		eqid 2761	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	clnbgrssvtx 48413	. . . . . . . 8 ⊢ (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺)
5	4	a1i 11	. . . . . . 7 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺))
6	3	isubgruhgr 48450	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ∈ UHGraph)
7	5, 6	sylan2 602	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ∈ UHGraph)
8		gricref 48502	. . . . . 6 ⊢ ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ∈ UHGraph → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))
9	7, 8	syl 17	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))
10	9	ralrimiva 3153	. . . 4 ⊢ (𝐺 ∈ UHGraph → ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))
11		f1oi 6839	. . . 4 ⊢ ( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺)
12	10, 11	jctil 527	. . 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 6788	. . . 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 6860	. . . . . . . . 9 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝑓‘𝑣) = (( I ↾ (Vtx‘𝐺))‘𝑣))
15	14	oveq2d 7406	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx (𝑓‘𝑣)) = (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣)))
16	15	oveq2d 7406	. . . . . . 7 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓‘𝑣))) = (𝐺 ISubGr (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣))))
17	16	breq2d 5109	. . . . . 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 7151	. . . . . . . . 9 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (( I ↾ (Vtx‘𝐺))‘𝑣) = 𝑣)
19	18	oveq2d 7406	. . . . . . . 8 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣)) = (𝐺 ClNeighbVtx 𝑣))
20	19	oveq2d 7406	. . . . . . 7 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (𝐺 ISubGr (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣))) = (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))
21	20	breq2d 5109	. . . . . 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 517	. . . . 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 3182	. . . 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 641	. . 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 3556	. 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 48590	. . 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 574	. 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 259	1 ⊢ (𝐺 ∈ UHGraph → 𝐺 ≃𝑙𝑔𝑟 𝐺)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ⊆ wss 3902   class class class wbr 5097   I cid 5537   ↾ cres 5645  –1-1-onto→wf1o 6514  ‘cfv 6515  (class class class)co 7390  Vtxcvtx 29153  UHGraphcuhgr 29213   ClNeighbVtx cclnbgr 48400   ISubGr cisubgr 48442   ≃_𝑔𝑟 cgric 48458   ≃_𝑙𝑔𝑟 cgrlic 48559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7964  df-2nd 7965  df-1o 8430  df-map 8803  df-vtx 29155  df-iedg 29156  df-uhgr 29215  df-clnbgr 48401  df-isubgr 48443  df-grim 48460  df-gric 48463  df-grlim 48560  df-grlic 48563

This theorem is referenced by:  grlicer  48598