Theorem reldmrelexp 14940

Description: The domain of the repeated composition of a relation is a relation. (Contributed by AV, 12-Jul-2024.)

Assertion

Ref	Expression
reldmrelexp	⊢ Rel dom ↑𝑟

Proof of Theorem reldmrelexp

Dummy variables 𝑛 𝑟 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-relexp 14939	. 2 ⊢ ↑𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))
2	1	reldmmpo 7517	1 ⊢ Rel dom ↑𝑟

Syntax hints:   = wceq 1541  Vcvv 3466   ∪ cun 3933  ifcif 4513   ↦ cmpt 5215   I cid 5557  dom cdm 5660  ran crn 5661   ↾ cres 5662   ∘ ccom 5664  Rel wrel 5665  ‘cfv 6523   ∈ cmpo 7386  0cc0 11082  1c1 11083  ℕ₀cn0 12444  seqcseq 13938  ↑_𝑟crelexp 14938

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5283  ax-nul 5290  ax-pr 5411

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3426  df-v 3468  df-dif 3938  df-un 3940  df-in 3942  df-ss 3952  df-nul 4310  df-if 4514  df-sn 4614  df-pr 4616  df-op 4620  df-br 5133  df-opab 5195  df-xp 5666  df-rel 5667  df-dm 5670  df-oprab 7388  df-mpo 7389  df-relexp 14939

This theorem is referenced by:  relexpsucrd  14952  relexpsucld  14953  relexpreld  14959  relexpdmd  14963  relexprnd  14967  relexpfldd  14969  relexpaddd  14973  dfrtrclrec2  14977  relexpindlem  14982