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Theorem grlimprop 47887
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:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣)))))
Distinct variable groups:   𝑣,𝐹   𝑣,𝐺   𝑣,𝐻   𝑣,𝑉
Allowed substitution hint:   𝑊(𝑣)

Proof of Theorem grlimprop
StepHypRef Expression
1 grlimdmrel 47883 . . . . 5 Rel dom GraphLocIso
21ovrcl 7472 . . . 4 (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V))
32simpld 494 . . 3 (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐺 ∈ V)
42simprd 495 . . 3 (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐻 ∈ V)
5 id 22 . . 3 (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
63, 4, 53jca 1127 . 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)
97, 8isgrlim 47885 . . 3 ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))))))
109biimpd 229 . 2 ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))))))
116, 10mpcom 38 1 (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478   class class class wbr 5148  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  Vtxcvtx 29028   ClNeighbVtx cclnbgr 47743   ISubGr cisubgr 47784  𝑔𝑟 cgric 47800   GraphLocIso cgrlim 47879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-grlim 47881
This theorem is referenced by:  grlimf1o  47888
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