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Theorem reldmrelexp 14968
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 14967 . 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 7543 1 Rel dom ↑𝑟
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  Vcvv 3475  cun 3947  ifcif 4529  cmpt 5232   I cid 5574  dom cdm 5677  ran crn 5678  cres 5679  ccom 5681  Rel wrel 5682  cfv 6544  cmpo 7411  0cc0 11110  1c1 11111  0cn0 12472  seqcseq 13966  𝑟crelexp 14966
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-dm 5687  df-oprab 7413  df-mpo 7414  df-relexp 14967
This theorem is referenced by:  relexpsucrd  14980  relexpsucld  14981  relexpreld  14987  relexpdmd  14991  relexprnd  14995  relexpfldd  14997  relexpaddd  15001  dfrtrclrec2  15005  relexpindlem  15010
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