Theorem grimdmrel 47819

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 47817	. 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 7536	1 ⊢ Rel dom GraphIso

Syntax hints:   ∧ wa 395   = wceq 1539  ∃wex 1778  {cab 2712  ∀wral 3050  Vcvv 3457  [wsbc 3763  dom cdm 5652   “ cima 5655  Rel wrel 5657  –1-1-onto→wf1o 6527  ‘cfv 6528  Vtxcvtx 28909  iEdgciedg 28910   GraphIso cgrim 47814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pr 5400

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-br 5118  df-opab 5180  df-xp 5658  df-rel 5659  df-dm 5662  df-oprab 7404  df-mpo 7405  df-grim 47817

This theorem is referenced by:  grimprop  47822  grimuhgr  47831  grimcnv  47832  grimco  47833  gricrcl  47836  uhgrimisgrgric  47852