Theorem grlimdmrel 48778

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 48776	. 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 7550	1 ⊢ Rel dom GraphLocIso

This proof depends on syntax axioms:   ∧ wa 401  {cab 2743  ∀wral 3081  Vcvv 3457   class class class wbr 5111  dom cdm 5663  Rel wrel 5668  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7416  Vtxcvtx 29377   ClNeighbVtx cclnbgr 48616   ISubGr cisubgr 48658   ≃_𝑔𝑟 cgric 48674   GraphLocIso cgrlim 48774

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-dm 5673  df-oprab 7420  df-mpo 7421  df-grlim 48776

This theorem is used by:  grlimprop  48782  grlimprop2  48784  grlicrcl  48805  grilcbri2  48809