Theorem reldmrelexp 15058

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 15057	. 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 7545	1 ⊢ Rel dom ↑𝑟

Syntax hints:   = wceq 1567  Vcvv 3463   ∪ cun 3911  ifcif 4492   ↦ cmpt 5196   I cid 5556  dom cdm 5662  ran crn 5663   ↾ cres 5664   ∘ ccom 5666  Rel wrel 5667  ‘cfv 6537   ∈ cmpo 7413  0cc0 11100  1c1 11101  ℕ₀cn0 12504  seqcseq 14037  ↑_𝑟crelexp 15056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-dm 5672  df-oprab 7415  df-mpo 7416  df-relexp 15057

This theorem is referenced by:  relexpsucrd  15070  relexpsucld  15071  relexpreld  15077  relexpdmd  15081  relexprnd  15085  relexpfldd  15087  relexpaddd  15091  dfrtrclrec2  15095  relexpindlem  15100