Theorem grlimdmrel 48334

Description: The domain of the graph local isomorphism function is a relation. (Contributed by AV, 20-May-2025.)

Assertion

Ref	Expression
grlimdmrel	⊢ Rel dom GraphLocIso

Proof of Theorem grlimdmrel

Dummy variables 𝑓 𝑔 ℎ 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-grlim 48332	. 2 ⊢ GraphLocIso = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (ℎ ISubGr (ℎ ClNeighbVtx (𝑓‘𝑣))))})
2	1	reldmmpo 7502	1 ⊢ Rel dom GraphLocIso

Syntax hints:   ∧ wa 395  {cab 2715  ∀wral 3052  Vcvv 3442   class class class wbr 5100  dom cdm 5632  Rel wrel 5637  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368  Vtxcvtx 29081   ClNeighbVtx cclnbgr 48172   ISubGr cisubgr 48214   ≃_𝑔𝑟 cgric 48230   GraphLocIso cgrlim 48330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-dm 5642  df-oprab 7372  df-mpo 7373  df-grlim 48332

This theorem is referenced by:  grlimprop  48338  grlimprop2  48340  grlicrcl  48361  grilcbri2  48365