Theorem grimdmrel 48686

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 48684	. 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 7557	1 ⊢ Rel dom GraphIso

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812  {cab 2744  ∀wral 3082  Vcvv 3458  [wsbc 3747  dom cdm 5666   “ cima 5669  Rel wrel 5671  –1-1-onto→wf1o 6542  ‘cfv 6543  Vtxcvtx 29383  iEdgciedg 29384   GraphIso cgrim 48681

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 2738  ax-sep 5262  ax-pr 5409

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-dm 5676  df-oprab 7427  df-mpo 7428  df-grim 48684

This theorem is used by:  grimprop  48689  grimuhgr  48693  grimcnv  48694  grimco  48695  gricrcl  48720  uhgrimisgrgric  48737