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Theorem isgrlim 47885
Description: A local isomorphism of graphs is a bijection between their vertices that preserves neighborhoods. (Contributed by AV, 20-May-2025.)
Hypotheses
Ref Expression
isgrlim.v 𝑉 = (Vtx‘𝐺)
isgrlim.w 𝑊 = (Vtx‘𝐻)
Assertion
Ref Expression
isgrlim ((𝐺𝑋𝐻𝑌𝐹𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))))))
Distinct variable groups:   𝑣,𝐹   𝑣,𝐺   𝑣,𝐻   𝑣,𝑉
Allowed substitution hints:   𝑊(𝑣)   𝑋(𝑣)   𝑌(𝑣)   𝑍(𝑣)

Proof of Theorem isgrlim
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-grlim 47881 . . 3 GraphLocIso = (𝑔 ∈ V, ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 ( ISubGr ( ClNeighbVtx (𝑓𝑣))))})
2 elex 3499 . . . 4 (𝐺𝑋𝐺 ∈ V)
323ad2ant1 1132 . . 3 ((𝐺𝑋𝐻𝑌𝐹𝑍) → 𝐺 ∈ V)
4 elex 3499 . . . 4 (𝐻𝑌𝐻 ∈ V)
543ad2ant2 1133 . . 3 ((𝐺𝑋𝐻𝑌𝐹𝑍) → 𝐻 ∈ V)
6 f1of 6849 . . . . . 6 (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) → 𝑓:(Vtx‘𝐺)⟶(Vtx‘𝐻))
76adantr 480 . . . . 5 ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣)))) → 𝑓:(Vtx‘𝐺)⟶(Vtx‘𝐻))
87adantl 481 . . . 4 (((𝐺𝑋𝐻𝑌𝐹𝑍) ∧ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))))) → 𝑓:(Vtx‘𝐺)⟶(Vtx‘𝐻))
9 fvexd 6922 . . . 4 ((𝐺𝑋𝐻𝑌𝐹𝑍) → (Vtx‘𝐺) ∈ V)
10 fvexd 6922 . . . 4 ((𝐺𝑋𝐻𝑌𝐹𝑍) → (Vtx‘𝐻) ∈ V)
118, 9, 10fabexd 7958 . . 3 ((𝐺𝑋𝐻𝑌𝐹𝑍) → {𝑓 ∣ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))))} ∈ V)
12 eqidd 2736 . . . . . 6 ((𝑔 = 𝐺 = 𝐻) → 𝑓 = 𝑓)
13 fveq2 6907 . . . . . . 7 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
1413adantr 480 . . . . . 6 ((𝑔 = 𝐺 = 𝐻) → (Vtx‘𝑔) = (Vtx‘𝐺))
15 fveq2 6907 . . . . . . 7 ( = 𝐻 → (Vtx‘) = (Vtx‘𝐻))
1615adantl 481 . . . . . 6 ((𝑔 = 𝐺 = 𝐻) → (Vtx‘) = (Vtx‘𝐻))
1712, 14, 16f1oeq123d 6843 . . . . 5 ((𝑔 = 𝐺 = 𝐻) → (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘) ↔ 𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)))
18 id 22 . . . . . . . 8 (𝑔 = 𝐺𝑔 = 𝐺)
19 oveq1 7438 . . . . . . . 8 (𝑔 = 𝐺 → (𝑔 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx 𝑣))
2018, 19oveq12d 7449 . . . . . . 7 (𝑔 = 𝐺 → (𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) = (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)))
21 id 22 . . . . . . . 8 ( = 𝐻 = 𝐻)
22 oveq1 7438 . . . . . . . 8 ( = 𝐻 → ( ClNeighbVtx (𝑓𝑣)) = (𝐻 ClNeighbVtx (𝑓𝑣)))
2321, 22oveq12d 7449 . . . . . . 7 ( = 𝐻 → ( ISubGr ( ClNeighbVtx (𝑓𝑣))) = (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))))
2420, 23breqan12d 5164 . . . . . 6 ((𝑔 = 𝐺 = 𝐻) → ((𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 ( ISubGr ( ClNeighbVtx (𝑓𝑣))) ↔ (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣)))))
2514, 24raleqbidv 3344 . . . . 5 ((𝑔 = 𝐺 = 𝐻) → (∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 ( ISubGr ( ClNeighbVtx (𝑓𝑣))) ↔ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣)))))
2617, 25anbi12d 632 . . . 4 ((𝑔 = 𝐺 = 𝐻) → ((𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 ( ISubGr ( ClNeighbVtx (𝑓𝑣)))) ↔ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))))))
2726abbidv 2806 . . 3 ((𝑔 = 𝐺 = 𝐻) → {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 ( ISubGr ( ClNeighbVtx (𝑓𝑣))))} = {𝑓 ∣ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))))})
281, 3, 5, 11, 27elovmpod 7677 . 2 ((𝐺𝑋𝐻𝑌𝐹𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))))}))
29 f1oeq1 6837 . . . . . 6 (𝑓 = 𝐹 → (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ↔ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)))
30 fveq1 6906 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑣) = (𝐹𝑣))
3130oveq2d 7447 . . . . . . . . 9 (𝑓 = 𝐹 → (𝐻 ClNeighbVtx (𝑓𝑣)) = (𝐻 ClNeighbVtx (𝐹𝑣)))
3231oveq2d 7447 . . . . . . . 8 (𝑓 = 𝐹 → (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))) = (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))))
3332breq2d 5160 . . . . . . 7 (𝑓 = 𝐹 → ((𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))) ↔ (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣)))))
3433ralbidv 3176 . . . . . 6 (𝑓 = 𝐹 → (∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))) ↔ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣)))))
3529, 34anbi12d 632 . . . . 5 (𝑓 = 𝐹 → ((𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣)))) ↔ (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))))))
3635elabg 3677 . . . 4 (𝐹𝑍 → (𝐹 ∈ {𝑓 ∣ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))))} ↔ (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))))))
37363ad2ant3 1134 . . 3 ((𝐺𝑋𝐻𝑌𝐹𝑍) → (𝐹 ∈ {𝑓 ∣ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))))} ↔ (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))))))
38 isgrlim.v . . . . 5 𝑉 = (Vtx‘𝐺)
39 isgrlim.w . . . . 5 𝑊 = (Vtx‘𝐻)
40 f1oeq23 6840 . . . . 5 ((𝑉 = (Vtx‘𝐺) ∧ 𝑊 = (Vtx‘𝐻)) → (𝐹:𝑉1-1-onto𝑊𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)))
4138, 39, 40mp2an 692 . . . 4 (𝐹:𝑉1-1-onto𝑊𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻))
4238raleqi 3322 . . . 4 (∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))) ↔ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))))
4341, 42anbi12i 628 . . 3 ((𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣)))) ↔ (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣)))))
4437, 43bitr4di 289 . 2 ((𝐺𝑋𝐻𝑌𝐹𝑍) → (𝐹 ∈ {𝑓 ∣ (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)(𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓𝑣))))} ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∀𝑣𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹𝑣))))))
4528, 44bitrd 279 1 ((𝐺𝑋𝐻𝑌𝐹𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (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  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wral 3059  Vcvv 3478   class class class wbr 5148  wf 6559  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  Vtxcvtx 29028   ClNeighbVtx cclnbgr 47743   ISubGr cisubgr 47784  𝑔𝑟 cgric 47800   GraphLocIso cgrlim 47879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-grlim 47881
This theorem is referenced by:  isgrlim2  47886  grlimprop  47887  uhgrimgrlim  47890  dfgrlic2  47904
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