Theorem uhgrimgrlim 48485

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 2740	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2740	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
3	1, 2	grimf1o 48382	. . 3 ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
4	3	3ad2ant3 1141	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
5		simpl1 1198	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → 𝐺 ∈ UHGraph)
6		simpl3 1200	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → 𝐹 ∈ (𝐺 GraphIso 𝐻))
7	1	clnbgrssvtx 48329	. . . . . 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 48429	. . . . 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 1379	. . . 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 1094	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ↔ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)))
12	1	clnbgrgrim 48432	. . . . . 6 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐻 ClNeighbVtx (𝐹‘𝑣)) = (𝐹 “ (𝐺 ClNeighbVtx 𝑣)))
13	11, 12	sylanb 587	. . . . 5 ⊢ (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐻 ClNeighbVtx (𝐹‘𝑣)) = (𝐹 “ (𝐺 ClNeighbVtx 𝑣)))
14	13	oveq2d 7379	. . . 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 5107	. . 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 3132	. 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 48480	. 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 719	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054   ⊆ wss 3890   class class class wbr 5079   “ cima 5628  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7363  Vtxcvtx 29090  UHGraphcuhgr 29150   ClNeighbVtx cclnbgr 48316   ISubGr cisubgr 48358   GraphIso cgrim 48373   ≃_𝑔𝑟 cgric 48374   GraphLocIso cgrlim 48474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-1o 8402  df-map 8772  df-vtx 29092  df-iedg 29093  df-edg 29142  df-uhgr 29152  df-clnbgr 48317  df-isubgr 48359  df-grim 48376  df-gric 48379  df-grlim 48476

This theorem is referenced by:  gricgrlic  48516