Theorem reldmrelexp 14928

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 14927	. 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 7483	1 ⊢ Rel dom ↑𝑟

Syntax hints:   = wceq 1540  Vcvv 3436   ∪ cun 3901  ifcif 4476   ↦ cmpt 5173   I cid 5513  dom cdm 5619  ran crn 5620   ↾ cres 5621   ∘ ccom 5623  Rel wrel 5624  ‘cfv 6482   ∈ cmpo 7351  0cc0 11009  1c1 11010  ℕ₀cn0 12384  seqcseq 13908  ↑_𝑟crelexp 14926

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 5235  ax-nul 5245  ax-pr 5371

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 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-dm 5629  df-oprab 7353  df-mpo 7354  df-relexp 14927

This theorem is referenced by:  relexpsucrd  14940  relexpsucld  14941  relexpreld  14947  relexpdmd  14951  relexprnd  14955  relexpfldd  14957  relexpaddd  14961  dfrtrclrec2  14965  relexpindlem  14970