Theorem grlimprop 48908

Description: Properties of a local isomorphism of graphs. (Contributed by AV, 21-May-2025.)

Hypotheses

Ref	Expression
grlimprop.v	⊢ 𝑉 = (Vtx‘𝐺)
grlimprop.w	⊢ 𝑊 = (Vtx‘𝐻)

Assertion

Ref	Expression
grlimprop	⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣)))))

Distinct variable groups:   𝑣,𝐹   𝑣,𝐺   𝑣,𝐻   𝑣,𝑉

Allowed substitution hint:   𝑊(𝑣)

Proof of Theorem grlimprop

Step	Hyp	Ref	Expression
1		grlimdmrel 48904	. . . . 5 ⊢ Rel dom GraphLocIso
2	1	ovrcl 7458	. . . 4 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V))
3	2	simpld 500	. . 3 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐺 ∈ V)
4	2	simprd 501	. . 3 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐻 ∈ V)
5		id 23	. . 3 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
6	3, 4, 5	3jca 1146	. 2 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)))
7		grlimprop.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
8		grlimprop.w	. . . 4 ⊢ 𝑊 = (Vtx‘𝐻)
9	7, 8	isgrlim 48906	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))))))
10	9	biimpd 232	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))))))
11	6, 10	mpcom 39	1 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   class class class wbr 5107  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7417  Vtxcvtx 29461   ClNeighbVtx cclnbgr 48742   ISubGr cisubgr 48784   ≃_𝑔𝑟 cgric 48800   GraphLocIso cgrlim 48900

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-grlim 48902

This theorem is used by:  grlimf1o  48909