Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isgrlim Structured version   Visualization version   GIF version

Theorem isgrlim 47949
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 47945 . . 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 3501 . . . 4 (𝐺𝑋𝐺 ∈ V)
323ad2ant1 1134 . . 3 ((𝐺𝑋𝐻𝑌𝐹𝑍) → 𝐺 ∈ V)
4 elex 3501 . . . 4 (𝐻𝑌𝐻 ∈ V)
543ad2ant2 1135 . . 3 ((𝐺𝑋𝐻𝑌𝐹𝑍) → 𝐻 ∈ V)
6 f1of 6848 . . . . . 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 6921 . . . 4 ((𝐺𝑋𝐻𝑌𝐹𝑍) → (Vtx‘𝐺) ∈ V)
10 fvexd 6921 . . . 4 ((𝐺𝑋𝐻𝑌𝐹𝑍) → (Vtx‘𝐻) ∈ V)
118, 9, 10fabexd 7959 . . 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 2738 . . . . . 6 ((𝑔 = 𝐺 = 𝐻) → 𝑓 = 𝑓)
13 fveq2 6906 . . . . . . 7 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
1413adantr 480 . . . . . 6 ((𝑔 = 𝐺 = 𝐻) → (Vtx‘𝑔) = (Vtx‘𝐺))
15 fveq2 6906 . . . . . . 7 ( = 𝐻 → (Vtx‘) = (Vtx‘𝐻))
1615adantl 481 . . . . . 6 ((𝑔 = 𝐺 = 𝐻) → (Vtx‘) = (Vtx‘𝐻))
1712, 14, 16f1oeq123d 6842 . . . . 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 5159 . . . . . 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 3346 . . . . 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 2808 . . 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 6836 . . . . . 6 (𝑓 = 𝐹 → (𝑓:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ↔ 𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻)))
30 fveq1 6905 . . . . . . . . . 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 5155 . . . . . . 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 3178 . . . . . 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 3676 . . . 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 1136 . . 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 6839 . . . . 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 3324 . . . 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 1087   = wceq 1540  wcel 2108  {cab 2714  wral 3061  Vcvv 3480   class class class wbr 5143  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Vtxcvtx 29013   ClNeighbVtx cclnbgr 47805   ISubGr cisubgr 47846  𝑔𝑟 cgric 47862   GraphLocIso cgrlim 47943
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-grlim 47945
This theorem is referenced by:  isgrlim2  47950  grlimprop  47951  uhgrimgrlim  47954  dfgrlic2  47968
  Copyright terms: Public domain W3C validator