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Theorem usgrlimprop 47933
Description: Properties of a local isomorphism of simple pseudographs. (Contributed by AV, 17-Aug-2025.)
Hypotheses
Ref Expression
usgrlimprop.v 𝑉 = (Vtx‘𝐺)
usgrlimprop.w 𝑊 = (Vtx‘𝐻)
usgrlimprop.n 𝑁 = (𝐺 ClNeighbVtx 𝑣)
usgrlimprop.m 𝑀 = (𝐻 ClNeighbVtx (𝐹𝑣))
usgrlimprop.i 𝐼 = (Edg‘𝐺)
usgrlimprop.j 𝐽 = (Edg‘𝐻)
usgrlimprop.k 𝐾 = {𝑥𝐼𝑥𝑁}
usgrlimprop.l 𝐿 = {𝑥𝐽𝑥𝑀}
Assertion
Ref Expression
usgrlimprop ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒)))))
Distinct variable groups:   𝑓,𝐹,𝑣   𝑒,𝐺,𝑓,𝑔,𝑣,𝑥   𝑒,𝐻,𝑓,𝑔,𝑣,𝑥   𝑥,𝐼   𝑥,𝐽   𝑒,𝐾,𝑔,𝑥   𝑔,𝐿,𝑥   𝑒,𝑀,𝑓,𝑔,𝑥   𝑒,𝑁,𝑓,𝑔,𝑥   𝑣,𝑉
Allowed substitution hints:   𝐹(𝑥,𝑒,𝑔)   𝐼(𝑣,𝑒,𝑓,𝑔)   𝐽(𝑣,𝑒,𝑓,𝑔)   𝐾(𝑣,𝑓)   𝐿(𝑣,𝑒,𝑓)   𝑀(𝑣)   𝑁(𝑣)   𝑉(𝑥,𝑒,𝑓,𝑔)   𝑊(𝑥,𝑣,𝑒,𝑓,𝑔)

Proof of Theorem usgrlimprop
StepHypRef Expression
1 simp3 1139 . 2 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
2 usgrlimprop.v . . 3 𝑉 = (Vtx‘𝐺)
3 usgrlimprop.w . . 3 𝑊 = (Vtx‘𝐻)
4 usgrlimprop.n . . 3 𝑁 = (𝐺 ClNeighbVtx 𝑣)
5 usgrlimprop.m . . 3 𝑀 = (𝐻 ClNeighbVtx (𝐹𝑣))
6 usgrlimprop.i . . 3 𝐼 = (Edg‘𝐺)
7 usgrlimprop.j . . 3 𝐽 = (Edg‘𝐻)
8 usgrlimprop.k . . 3 𝐾 = {𝑥𝐼𝑥𝑁}
9 usgrlimprop.l . . 3 𝐿 = {𝑥𝐽𝑥𝑀}
102, 3, 4, 5, 6, 7, 8, 9uspgrlim 47932 . 2 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒))))))
111, 10mpbid 232 1 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑒𝐾 (𝑓𝑒) = (𝑔𝑒)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wral 3060  {crab 3435  wss 3950  cima 5686  1-1-ontowf1o 6558  cfv 6559  (class class class)co 7429  Vtxcvtx 29003  Edgcedg 29054  USPGraphcuspgr 29155   ClNeighbVtx cclnbgr 47778   GraphLocIso cgrlim 47916
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-suc 6388  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-1o 8502  df-map 8864  df-vtx 29005  df-iedg 29006  df-edg 29055  df-uspgr 29157  df-clnbgr 47779  df-isubgr 47820  df-grim 47837  df-gric 47840  df-grlim 47918
This theorem is referenced by:  grlimgrtri  47936
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