Theorem reldmrelexp 14413

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 14412	. 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 7273	1 ⊢ Rel dom ↑𝑟

Syntax hints:   = wceq 1539  Vcvv 3407   ∪ cun 3852  ifcif 4413   ↦ cmpt 5105   I cid 5422  dom cdm 5517  ran crn 5518   ↾ cres 5519   ∘ ccom 5521  Rel wrel 5522  ‘cfv 6328   ∈ cmpo 7145  0cc0 10560  1c1 10561  ℕ₀cn0 11919  seqcseq 13403  ↑_𝑟crelexp 14411

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pr 5291

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-v 3409  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-nul 4222  df-if 4414  df-sn 4516  df-pr 4518  df-op 4522  df-br 5026  df-opab 5088  df-xp 5523  df-rel 5524  df-dm 5527  df-oprab 7147  df-mpo 7148  df-relexp 14412

This theorem is referenced by:  relexpsucrd  14425  relexpsucld  14426  relexpreld  14432  relexpdmd  14436  relexprnd  14440  relexpfldd  14442  relexpaddd  14446  dfrtrclrec2  14450  relexpindlem  14455