Theorem grlicref 47538

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 6908	. . . 4 ⊢ (𝐺 ∈ UHGraph → (Vtx‘𝐺) ∈ V)
2	1	resiexd 7225	. . 3 ⊢ (𝐺 ∈ UHGraph → ( I ↾ (Vtx‘𝐺)) ∈ V)
3		eqid 2726	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	clnbgrssvtx 47438	. . . . . . . 8 ⊢ (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺)
5	4	a1i 11	. . . . . . 7 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺))
6	3	isubgruhgr 47469	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ∈ UHGraph)
7	5, 6	sylan2 591	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ∈ UHGraph)
8		gricref 47504	. . . . . 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 3136	. . . 4 ⊢ (𝐺 ∈ UHGraph → ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))
11		f1oi 6873	. . . 4 ⊢ ( I ↾ (Vtx‘𝐺)):(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐺)
12	10, 11	jctil 518	. . 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 6823	. . . 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 6892	. . . . . . . . 9 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝑓‘𝑣) = (( I ↾ (Vtx‘𝐺))‘𝑣))
15	14	oveq2d 7432	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx (𝑓‘𝑣)) = (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣)))
16	15	oveq2d 7432	. . . . . . 7 ⊢ (𝑓 = ( I ↾ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx (𝑓‘𝑣))) = (𝐺 ISubGr (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣))))
17	16	breq2d 5157	. . . . . 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 7432	. . . . . . . 8 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣)) = (𝐺 ClNeighbVtx 𝑣))
20	19	oveq2d 7432	. . . . . . 7 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (𝐺 ISubGr (𝐺 ClNeighbVtx (( I ↾ (Vtx‘𝐺))‘𝑣))) = (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))
21	20	breq2d 5157	. . . . . 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 508	. . . . 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 3166	. . . 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 630	. . 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 3583	. 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 47534	. . 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 565	. 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 256	1 ⊢ (𝐺 ∈ UHGraph → 𝐺 ≃𝑙𝑔𝑟 𝐺)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534  ∃wex 1774   ∈ wcel 2099  ∀wral 3051  Vcvv 3462   ⊆ wss 3946   class class class wbr 5145   I cid 5571   ↾ cres 5676  –1-1-onto→wf1o 6545  ‘cfv 6546  (class class class)co 7416  Vtxcvtx 28929  UHGraphcuhgr 28989   ClNeighbVtx cclnbgr 47426   ISubGr cisubgr 47463   ≃_𝑔𝑟 cgric 47477   ≃_𝑙𝑔𝑟 cgrlic 47519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7995  df-2nd 7996  df-1o 8488  df-map 8849  df-vtx 28931  df-iedg 28932  df-uhgr 28991  df-clnbgr 47427  df-isubgr 47464  df-grim 47479  df-gric 47482  df-grlim 47520  df-grlic 47523

This theorem is referenced by:  grlicer  47542