Theorem reldmrelexp 15045

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 15044	. 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 7546	1 ⊢ Rel dom ↑𝑟

Syntax hints:   = wceq 1540  Vcvv 3464   ∪ cun 3929  ifcif 4505   ↦ cmpt 5206   I cid 5552  dom cdm 5659  ran crn 5660   ↾ cres 5661   ∘ ccom 5663  Rel wrel 5664  ‘cfv 6536   ∈ cmpo 7412  0cc0 11134  1c1 11135  ℕ₀cn0 12506  seqcseq 14024  ↑_𝑟crelexp 15043

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-dm 5669  df-oprab 7414  df-mpo 7415  df-relexp 15044

This theorem is referenced by:  relexpsucrd  15057  relexpsucld  15058  relexpreld  15064  relexpdmd  15068  relexprnd  15072  relexpfldd  15074  relexpaddd  15078  dfrtrclrec2  15082  relexpindlem  15087