Theorem uhgrimgrlim 49029

Description: An isomorphism of hypergraphs is a local isomorphism between the two graphs. (Contributed by AV, 2-Jun-2025.)

Assertion

Ref	Expression
uhgrimgrlim	⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))

Proof of Theorem uhgrimgrlim

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2761	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2761	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
3	1, 2	grimf1o 48926	. . 3 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
4	3	3ad2ant3 1153	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
5		simpl1 1210	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → 𝐺 ∈ UHGraph)
6		simpl3 1212	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → 𝐹 ∈ (𝐺 GraphIso 𝐻))
7	1	clnbgrssvtx 48873	. . . . . 6 ⊢ (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺)
8	7	a1i 11	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺))
9	1	uhgrimisgrgric 48973	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝐺 ClNeighbVtx 𝑣) ⊆ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐹 “ (𝐺 ClNeighbVtx 𝑣))))
10	5, 6, 8, 9	syl3anc 1398	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐹 “ (𝐺 ClNeighbVtx 𝑣))))
11		df-3an 1105	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ↔ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)))
12	1	clnbgrgrim 48976	. . . . . 6 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐻 ClNeighbVtx (𝐹‘𝑣)) = (𝐹 “ (𝐺 ClNeighbVtx 𝑣)))
13	11, 12	sylanb 593	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐻 ClNeighbVtx (𝐹‘𝑣)) = (𝐹 “ (𝐺 ClNeighbVtx 𝑣)))
14	13	oveq2d 7428	. . . 4 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))) = (𝐻 ISubGr (𝐹 “ (𝐺 ClNeighbVtx 𝑣))))
15	10, 14	breqtrrd 5133	. . 3 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))))
16	15	ralrimiva 3155	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))))
17	1, 2	isgrlim 49024	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))))))
18	4, 16, 17	mpbir2and 726	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ⊆ wss 3899   class class class wbr 5103   “ cima 5654  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7412  Vtxcvtx 29556  UHGraphcuhgr 29616   ClNeighbVtx cclnbgr 48860   ISubGr cisubgr 48902   GraphIso cgrim 48917   ≃_𝑔𝑟 cgric 48918   GraphLocIso cgrlim 49018

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  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 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-1o 8460  df-map 8833  df-vtx 29558  df-iedg 29559  df-edg 29608  df-uhgr 29618  df-clnbgr 48861  df-isubgr 48903  df-grim 48920  df-gric 48923  df-grlim 49020

This theorem is used by:  gricgrlic  49060