Theorem grlimdmrel 47965

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 47963	. 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 7487	1 ⊢ Rel dom GraphLocIso

Syntax hints:   ∧ wa 395  {cab 2707  ∀wral 3044  Vcvv 3438   class class class wbr 5095  dom cdm 5623  Rel wrel 5628  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353  Vtxcvtx 28959   ClNeighbVtx cclnbgr 47803   ISubGr cisubgr 47845   ≃_𝑔𝑟 cgric 47861   GraphLocIso cgrlim 47961

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-dm 5633  df-oprab 7357  df-mpo 7358  df-grlim 47963

This theorem is referenced by:  grlimprop  47969  grlimprop2  47971  grlicrcl  47992  grilcbri2  47996