Theorem grimdmrel 47868

Description: The domain of the graph isomorphism function is a relation. (Contributed by AV, 28-Apr-2025.)

Assertion

Ref	Expression
grimdmrel	⊢ Rel dom GraphIso

Proof of Theorem grimdmrel

Dummy variables 𝑒 𝑑 𝑓 𝑔 ℎ 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-grim 47866	. 2 ⊢ GraphIso = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∃𝑗[(iEdg‘𝑔) / 𝑒][(iEdg‘ℎ) / 𝑑](𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖))))})
2	1	reldmmpo 7487	1 ⊢ Rel dom GraphIso

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779  {cab 2707  ∀wral 3044  Vcvv 3438  [wsbc 3744  dom cdm 5623   “ cima 5626  Rel wrel 5628  –1-1-onto→wf1o 6485  ‘cfv 6486  Vtxcvtx 28959  iEdgciedg 28960   GraphIso cgrim 47863

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-grim 47866

This theorem is referenced by:  grimprop  47871  grimuhgr  47875  grimcnv  47876  grimco  47877  gricrcl  47902  uhgrimisgrgric  47919