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Theorem dfgrlic2 47968
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)
Assertion
Ref Expression
dfgrlic2 ((𝐺𝑋𝐻𝑌) → (𝐺𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))))))
Distinct variable groups:   𝑓,𝐺,𝑣   𝑓,𝐻,𝑣   𝑣,𝑉   𝑓,𝑋   𝑓,𝑌
Allowed substitution hints:   𝑉(𝑓)   𝑊(𝑣,𝑓)   𝑋(𝑣)   𝑌(𝑣)

Proof of Theorem dfgrlic2
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)
64, 5isgrlim 47949 . . . 4 ((𝐺𝑋𝐻𝑌𝑓 ∈ V) → (𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))))))
76el3v3 3489 . . 3 ((𝐺𝑋𝐻𝑌) → (𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))))))
87exbidv 1921 . 2 ((𝐺𝑋𝐻𝑌) → (∃𝑓 𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))))))
93, 8bitrid 283 1 ((𝐺𝑋𝐻𝑌) → (𝐺𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(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  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  Vcvv 3480  c0 4333   class class class wbr 5143  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Vtxcvtx 29013   ClNeighbVtx cclnbgr 47805   ISubGr cisubgr 47846  𝑔𝑟 cgric 47862   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-grlim 47945  df-grlic 47948
This theorem is referenced by:  grilcbri  47969  grlicref  47972  grlicsym  47973  grlictr  47975  clnbgr3stgrgrlic  47979
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