Theorem grimdmrel 48368

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 48366	. 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 7494	1 ⊢ Rel dom GraphIso

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781  {cab 2715  ∀wral 3052  Vcvv 3430  [wsbc 3729  dom cdm 5624   “ cima 5627  Rel wrel 5629  –1-1-onto→wf1o 6491  ‘cfv 6492  Vtxcvtx 29079  iEdgciedg 29080   GraphIso cgrim 48363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-dm 5634  df-oprab 7364  df-mpo 7365  df-grim 48366

This theorem is referenced by:  grimprop  48371  grimuhgr  48375  grimcnv  48376  grimco  48377  gricrcl  48402  uhgrimisgrgric  48419