Theorem grimdmrel 47880

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 47878	. 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 7523	1 ⊢ Rel dom GraphIso

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779  {cab 2707  ∀wral 3044  Vcvv 3447  [wsbc 3753  dom cdm 5638   “ cima 5641  Rel wrel 5643  –1-1-onto→wf1o 6510  ‘cfv 6511  Vtxcvtx 28923  iEdgciedg 28924   GraphIso cgrim 47875

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 5251  ax-nul 5261  ax-pr 5387

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-dm 5648  df-oprab 7391  df-mpo 7392  df-grim 47878

This theorem is referenced by:  grimprop  47883  grimuhgr  47887  grimcnv  47888  grimco  47889  gricrcl  47914  uhgrimisgrgric  47931