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Theorem reldmrelexp 14389
 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.
StepHypRef Expression
1 df-relexp 14388 . 2 𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))
21reldmmpo 7272 1 Rel dom ↑𝑟
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  Vcvv 3441   ∪ cun 3880  ifcif 4427   ↦ cmpt 5113   I cid 5427  dom cdm 5522  ran crn 5523   ↾ cres 5524   ∘ ccom 5526  Rel wrel 5527  ‘cfv 6329   ∈ cmpo 7144  0cc0 10541  1c1 10542  ℕ0cn0 11900  seqcseq 13381  ↑𝑟crelexp 14387 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-sn 4528  df-pr 4530  df-op 4534  df-br 5034  df-opab 5096  df-xp 5528  df-rel 5529  df-dm 5532  df-oprab 7146  df-mpo 7147  df-relexp 14388 This theorem is referenced by:  relexpsucrd  14401  relexpsucld  14402  relexpreld  14408  relexpdmd  14412  relexprnd  14416  relexpfldd  14418  relexpaddd  14422  dfrtrclrec2  14426  relexpindlem  14431
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