Theorem uhgrimgrlim 47966

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 2736	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2736	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
3	1, 2	grimf1o 47864	. . 3 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
4	3	3ad2ant3 1135	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
5		simpl1 1192	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → 𝐺 ∈ UHGraph)
6		simpl3 1194	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → 𝐹 ∈ (𝐺 GraphIso 𝐻))
7	1	clnbgrssvtx 47812	. . . . . 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 47911	. . . . 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 1373	. . . 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 1088	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ↔ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)))
12	1	clnbgrgrim 47914	. . . . . 6 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐻 ClNeighbVtx (𝐹‘𝑣)) = (𝐹 “ (𝐺 ClNeighbVtx 𝑣)))
13	11, 12	sylanb 581	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐻 ClNeighbVtx (𝐹‘𝑣)) = (𝐹 “ (𝐺 ClNeighbVtx 𝑣)))
14	13	oveq2d 7426	. . . 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 5152	. . 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 3133	. 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 47961	. 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 713	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052   ⊆ wss 3931   class class class wbr 5124   “ cima 5662  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  Vtxcvtx 28980  UHGraphcuhgr 29040   ClNeighbVtx cclnbgr 47799   ISubGr cisubgr 47840   GraphIso cgrim 47855   ≃_𝑔𝑟 cgric 47856   GraphLocIso cgrlim 47955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-1o 8485  df-map 8847  df-vtx 28982  df-iedg 28983  df-edg 29032  df-uhgr 29042  df-clnbgr 47800  df-isubgr 47841  df-grim 47858  df-gric 47861  df-grlim 47957

This theorem is referenced by:  gricgrlic  47990