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Theorem dfgrlic3 47905
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 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)
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 47885 . . . 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 3486 . . 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 1918 . 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 1536  wex 1775  wcel 2105  wne 2937  wral 3058  {crab 3432  Vcvv 3477  wss 3962  c0 4338   class class class wbr 5147  dom cdm 5688  cima 5691  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  Vtxcvtx 29027  iEdgciedg 29028   ClNeighbVtx cclnbgr 47742   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-map 8866  df-vtx 29029  df-iedg 29030  df-clnbgr 47743  df-isubgr 47784  df-grim 47801  df-gric 47804  df-grlim 47880  df-grlic 47883
This theorem is referenced by:  grilcbri2  47906
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