Theorem grlimdmrel 48222

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 48220	. 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 7492	1 ⊢ Rel dom GraphLocIso

Syntax hints:   ∧ wa 395  {cab 2714  ∀wral 3051  Vcvv 3440   class class class wbr 5098  dom cdm 5624  Rel wrel 5629  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358  Vtxcvtx 29069   ClNeighbVtx cclnbgr 48060   ISubGr cisubgr 48102   ≃_𝑔𝑟 cgric 48118   GraphLocIso cgrlim 48218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-dm 5634  df-oprab 7362  df-mpo 7363  df-grlim 48220

This theorem is referenced by:  grlimprop  48226  grlimprop2  48228  grlicrcl  48249  grilcbri2  48253