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Theorem dfgrlic3 47970
Description: Alternate, explicit definition of the "is locally isomorphic to" relation for two graphs. (Contributed by AV, 9-Jun-2025.)
Hypotheses
Ref Expression
dfgrlic2.v 𝑉 = (Vtx‘𝐺)
dfgrlic2.w 𝑊 = (Vtx‘𝐻)
dfgrlic3.i 𝐼 = (iEdg‘𝐺)
dfgrlic3.j 𝐽 = (iEdg‘𝐻)
dfgrlic3.n 𝑁 = (𝐺 ClNeighbVtx 𝑣)
dfgrlic3.m 𝑀 = (𝐻 ClNeighbVtx (𝑓𝑣))
dfgrlic3.k 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) ⊆ 𝑁}
dfgrlic3.l 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) ⊆ 𝑀}
Assertion
Ref Expression
dfgrlic3 ((𝐺𝑋𝐻𝑌) → (𝐺𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑗(𝑗:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑗 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))))
Distinct variable groups:   𝑓,𝐺,𝑣   𝑓,𝐻,𝑣   𝑣,𝑉   𝑓,𝑋   𝑓,𝑌   𝑔,𝐺,𝑖,𝑗,𝑓,𝑣   𝑥,𝐺   𝑔,𝐻,𝑖,𝑗   𝑥,𝐻   𝑖,𝐼,𝑥   𝑖,𝐽,𝑥   𝑖,𝐾   𝑔,𝑋,𝑗,𝑣   𝑖,𝐿   𝑔,𝑀,𝑖,𝑗   𝑥,𝑀   𝑔,𝑁,𝑖,𝑗   𝑥,𝑁   𝑖,𝑋   𝑖,𝑌,𝑗,𝑔,𝑣
Allowed substitution hints:   𝐼(𝑣,𝑓,𝑔,𝑗)   𝐽(𝑣,𝑓,𝑔,𝑗)   𝐾(𝑥,𝑣,𝑓,𝑔,𝑗)   𝐿(𝑥,𝑣,𝑓,𝑔,𝑗)   𝑀(𝑣,𝑓)   𝑁(𝑣,𝑓)   𝑉(𝑥,𝑓,𝑔,𝑖,𝑗)   𝑊(𝑥,𝑣,𝑓,𝑔,𝑖,𝑗)   𝑋(𝑥)   𝑌(𝑥)

Proof of Theorem dfgrlic3
StepHypRef Expression
1 brgrlic 47964 . . 3 (𝐺𝑙𝑔𝑟 𝐻 ↔ (𝐺 GraphLocIso 𝐻) ≠ ∅)
2 n0 4353 . . 3 ((𝐺 GraphLocIso 𝐻) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐺 GraphLocIso 𝐻))
31, 2bitri 275 . 2 (𝐺𝑙𝑔𝑟 𝐻 ↔ ∃𝑓 𝑓 ∈ (𝐺 GraphLocIso 𝐻))
4 dfgrlic2.v . . . . 5 𝑉 = (Vtx‘𝐺)
5 dfgrlic2.w . . . . 5 𝑊 = (Vtx‘𝐻)
6 dfgrlic3.n . . . . 5 𝑁 = (𝐺 ClNeighbVtx 𝑣)
7 dfgrlic3.m . . . . 5 𝑀 = (𝐻 ClNeighbVtx (𝑓𝑣))
8 dfgrlic3.i . . . . 5 𝐼 = (iEdg‘𝐺)
9 dfgrlic3.j . . . . 5 𝐽 = (iEdg‘𝐻)
10 dfgrlic3.k . . . . 5 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) ⊆ 𝑁}
11 dfgrlic3.l . . . . 5 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) ⊆ 𝑀}
124, 5, 6, 7, 8, 9, 10, 11isgrlim2 47950 . . . 4 ((𝐺𝑋𝐻𝑌𝑓 ∈ V) → (𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑗(𝑗:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑗 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))))
1312el3v3 3489 . . 3 ((𝐺𝑋𝐻𝑌) → (𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑗(𝑗:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑗 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))))
1413exbidv 1921 . 2 ((𝐺𝑋𝐻𝑌) → (∃𝑓 𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑗(𝑗:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑗 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))))
153, 14bitrid 283 1 ((𝐺𝑋𝐻𝑌) → (𝐺𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑗(𝑗:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑗 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  {crab 3436  Vcvv 3480  wss 3951  c0 4333   class class class wbr 5143  dom cdm 5685  cima 5688  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Vtxcvtx 29013  iEdgciedg 29014   ClNeighbVtx cclnbgr 47805   GraphLocIso cgrlim 47943  𝑙𝑔𝑟 cgrlic 47944
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 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
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 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-map 8868  df-vtx 29015  df-iedg 29016  df-clnbgr 47806  df-isubgr 47847  df-grim 47864  df-gric 47867  df-grlim 47945  df-grlic 47948
This theorem is referenced by:  grilcbri2  47971
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