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Theorem isgrlim2 47981
Description: A local isomorphism of graphs is a bijection between their vertices that preserves neighborhoods. Definitions expanded. (Contributed by AV, 29-May-2025.)
Hypotheses
Ref Expression
isgrlim.v 𝑉 = (Vtx‘𝐺)
isgrlim.w 𝑊 = (Vtx‘𝐻)
isgrlim2.n 𝑁 = (𝐺 ClNeighbVtx 𝑣)
isgrlim2.m 𝑀 = (𝐻 ClNeighbVtx (𝐹𝑣))
isgrlim2.i 𝐼 = (iEdg‘𝐺)
isgrlim2.j 𝐽 = (iEdg‘𝐻)
isgrlim2.k 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) ⊆ 𝑁}
isgrlim2.l 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) ⊆ 𝑀}
Assertion
Ref Expression
isgrlim2 ((𝐺𝑋𝐻𝑌𝐹𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))))
Distinct variable groups:   𝑓,𝐹,𝑔,𝑣   𝑓,𝐺,𝑔,𝑣   𝑓,𝐻,𝑔,𝑣   𝑣,𝑉   𝑓,𝑋,𝑔   𝑓,𝑌,𝑔   𝑓,𝑍,𝑔   𝑖,𝐺   𝑥,𝐺   𝑖,𝐻   𝑥,𝐻   𝑥,𝐼   𝑥,𝐽   𝑖,𝐾   𝑖,𝐿   𝑓,𝑀,𝑔,𝑖   𝑥,𝑀   𝑓,𝑁,𝑔,𝑖   𝑥,𝑁   𝑖,𝑋,𝑣   𝑖,𝑌,𝑣   𝑣,𝑍
Allowed substitution hints:   𝐹(𝑥,𝑖)   𝐼(𝑣,𝑓,𝑔,𝑖)   𝐽(𝑣,𝑓,𝑔,𝑖)   𝐾(𝑥,𝑣,𝑓,𝑔)   𝐿(𝑥,𝑣,𝑓,𝑔)   𝑀(𝑣)   𝑁(𝑣)   𝑉(𝑥,𝑓,𝑔,𝑖)   𝑊(𝑥,𝑣,𝑓,𝑔,𝑖)   𝑋(𝑥)   𝑌(𝑥)   𝑍(𝑥,𝑖)

Proof of Theorem isgrlim2
StepHypRef Expression
1 isgrlim.v . . 3 𝑉 = (Vtx‘𝐺)
2 isgrlim.w . . 3 𝑊 = (Vtx‘𝐻)
31, 2isgrlim 47980 . 2 ((𝐺𝑋𝐻𝑌𝐹𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))))))
4 isgrlim2.n . . . . . . . . 9 𝑁 = (𝐺 ClNeighbVtx 𝑣)
54eqcomi 2738 . . . . . . . 8 (𝐺 ClNeighbVtx 𝑣) = 𝑁
65oveq2i 7351 . . . . . . 7 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) = (𝐺 ISubGr 𝑁)
7 isgrlim2.m . . . . . . . . 9 𝑀 = (𝐻 ClNeighbVtx (𝐹𝑣))
87eqcomi 2738 . . . . . . . 8 (𝐻 ClNeighbVtx (𝐹𝑣)) = 𝑀
98oveq2i 7351 . . . . . . 7 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))) = (𝐻 ISubGr 𝑀)
106, 9breq12i 5097 . . . . . 6 ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))) ↔ (𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr 𝑀))
1110a1i 11 . . . . 5 ((𝐺𝑋𝐻𝑌𝐹𝑍) → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))) ↔ (𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr 𝑀)))
12 isgrlim2.i . . . . . . 7 𝐼 = (iEdg‘𝐺)
13 isgrlim2.j . . . . . . 7 𝐽 = (iEdg‘𝐻)
14 isgrlim2.k . . . . . . 7 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼𝑥) ⊆ 𝑁}
15 isgrlim2.l . . . . . . 7 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽𝑥) ⊆ 𝑀}
1612, 13, 4, 7, 14, 15clnbgrisubgrgrim 47930 . . . . . 6 ((𝐺𝑋𝐻𝑌) → ((𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr 𝑀) ↔ ∃𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
17163adant3 1132 . . . . 5 ((𝐺𝑋𝐻𝑌𝐹𝑍) → ((𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr 𝑀) ↔ ∃𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
1811, 17bitrd 279 . . . 4 ((𝐺𝑋𝐻𝑌𝐹𝑍) → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))) ↔ ∃𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
1918ralbidv 3152 . . 3 ((𝐺𝑋𝐻𝑌𝐹𝑍) → (∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))) ↔ ∀𝑣𝑉𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))
2019anbi2d 630 . 2 ((𝐺𝑋𝐻𝑌𝐹𝑍) → ((𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣)))) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))))
213, 20bitrd 279 1 ((𝐺𝑋𝐻𝑌𝐹𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉𝑓(𝑓:𝑁1-1-onto𝑀 ∧ ∃𝑔(𝑔:𝐾1-1-onto𝐿 ∧ ∀𝑖𝐾 (𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1540  wex 1779  wcel 2109  wral 3044  {crab 3392  wss 3899   class class class wbr 5088  dom cdm 5613  cima 5616  1-1-ontowf1o 6475  cfv 6476  (class class class)co 7340  Vtxcvtx 28928  iEdgciedg 28929   ClNeighbVtx cclnbgr 47816   ISubGr cisubgr 47858  𝑔𝑟 cgric 47874   GraphLocIso cgrlim 47974
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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7343  df-oprab 7344  df-mpo 7345  df-1st 7915  df-2nd 7916  df-1o 8379  df-map 8746  df-vtx 28930  df-iedg 28931  df-clnbgr 47817  df-isubgr 47859  df-grim 47876  df-gric 47879  df-grlim 47976
This theorem is referenced by:  grlimprop2  47984  uspgrlim  47990  dfgrlic3  48008
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