Theorem grlimdmrel 48728

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 48726	. 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 7546	1 ⊢ Rel dom GraphLocIso

Syntax hints:   ∧ wa 400  {cab 2741  ∀wral 3079  Vcvv 3455   class class class wbr 5110  dom cdm 5663  Rel wrel 5668  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7412  Vtxcvtx 29327   ClNeighbVtx cclnbgr 48566   ISubGr cisubgr 48608   ≃_𝑔𝑟 cgric 48624   GraphLocIso cgrlim 48724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-dm 5673  df-oprab 7416  df-mpo 7417  df-grlim 48726

This theorem is referenced by:  grlimprop  48732  grlimprop2  48734  grlicrcl  48755  grilcbri2  48759