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Theorem usgrlimprop 47818
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 1136 . 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 47817 . 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 1085   = wceq 1535  wex 1774  wcel 2104  wral 3057  {crab 3432  wss 3963  cima 5686  1-1-ontowf1o 6557  cfv 6558  (class class class)co 7425  Vtxcvtx 29009  Edgcedg 29060  USPGraphcuspgr 29161   ClNeighbVtx cclnbgr 47693   GraphLocIso cgrlim 47801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5366  ax-pr 5430  ax-un 7747
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  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 6386  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7428  df-oprab 7429  df-mpo 7430  df-1st 8007  df-2nd 8008  df-1o 8499  df-map 8861  df-vtx 29011  df-iedg 29012  df-edg 29061  df-uspgr 29163  df-clnbgr 47694  df-isubgr 47734  df-grim 47749  df-gric 47752  df-grlim 47803
This theorem is referenced by:  grlimgrtri  47821
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