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Theorem dfgrlic2 47903
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 47899 . . 3 (𝐺𝑙𝑔𝑟 𝐻 ↔ (𝐺 GraphLocIso 𝐻) ≠ ∅)
2 n0 4358 . . 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 47884 . . . 4 ((𝐺𝑋𝐻𝑌𝑓 ∈ V) → (𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))))))
76el3v3 3486 . . 3 ((𝐺𝑋𝐻𝑌) → (𝑓 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝑓:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))))))
87exbidv 1918 . 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 1536  wex 1775  wcel 2105  wne 2937  wral 3058  Vcvv 3477  c0 4338   class class class wbr 5147  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  Vtxcvtx 29027   ClNeighbVtx cclnbgr 47742   ISubGr cisubgr 47783  𝑔𝑟 cgric 47799   GraphLocIso cgrlim 47878  𝑙𝑔𝑟 cgrlic 47879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-1o 8504  df-grlim 47880  df-grlic 47883
This theorem is referenced by:  grilcbri  47904  grlicref  47907  grlicsym  47908  grlictr  47910  clnbgr3stgrgrlic  47914
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