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Theorem reldmrelexp 14970
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 14969 . 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 7545 1 Rel dom ↑𝑟
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  Vcvv 3474  cun 3946  ifcif 4528  cmpt 5231   I cid 5573  dom cdm 5676  ran crn 5677  cres 5678  ccom 5680  Rel wrel 5681  cfv 6543  cmpo 7413  0cc0 11112  1c1 11113  0cn0 12474  seqcseq 13968  𝑟crelexp 14968
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-dm 5686  df-oprab 7415  df-mpo 7416  df-relexp 14969
This theorem is referenced by:  relexpsucrd  14982  relexpsucld  14983  relexpreld  14989  relexpdmd  14993  relexprnd  14997  relexpfldd  14999  relexpaddd  15003  dfrtrclrec2  15007  relexpindlem  15012
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