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Theorem dfgrlic3 47536
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 47530 . . 3 (𝐺𝑙𝑔𝑟 𝐻 ↔ (𝐺 GraphLocIso 𝐻) ≠ ∅)
2 n0 4346 . . 3 ((𝐺 GraphLocIso 𝐻) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐺 GraphLocIso 𝐻))
31, 2bitri 274 . 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 47525 . . . 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 3471 . . 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 1917 . 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 282 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 205  wa 394   = wceq 1534  wex 1774  wcel 2099  wne 2930  wral 3051  {crab 3419  Vcvv 3462  wss 3946  c0 4322   class class class wbr 5145  dom cdm 5674  cima 5677  1-1-ontowf1o 6545  cfv 6546  (class class class)co 7416  Vtxcvtx 28929  iEdgciedg 28930   ClNeighbVtx cclnbgr 47426   GraphLocIso cgrlim 47518  𝑙𝑔𝑟 cgrlic 47519
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 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7995  df-2nd 7996  df-1o 8488  df-map 8849  df-vtx 28931  df-iedg 28932  df-clnbgr 47427  df-isubgr 47464  df-grim 47479  df-gric 47482  df-grlim 47520  df-grlic 47523
This theorem is referenced by:  grilcbri2  47537
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