Theorem reldmrelexp 14732

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 14731	. 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 7408	1 ⊢ Rel dom ↑𝑟

Syntax hints:   = wceq 1539  Vcvv 3432   ∪ cun 3885  ifcif 4459   ↦ cmpt 5157   I cid 5488  dom cdm 5589  ran crn 5590   ↾ cres 5591   ∘ ccom 5593  Rel wrel 5594  ‘cfv 6433   ∈ cmpo 7277  0cc0 10871  1c1 10872  ℕ₀cn0 12233  seqcseq 13721  ↑_𝑟crelexp 14730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-dm 5599  df-oprab 7279  df-mpo 7280  df-relexp 14731

This theorem is referenced by:  relexpsucrd  14744  relexpsucld  14745  relexpreld  14751  relexpdmd  14755  relexprnd  14759  relexpfldd  14761  relexpaddd  14765  dfrtrclrec2  14769  relexpindlem  14774