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Theorem reldmrelexp 14898
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 14897 . 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 7486 1 Rel dom ↑𝑟
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  Vcvv 3443  cun 3906  ifcif 4484  cmpt 5186   I cid 5528  dom cdm 5631  ran crn 5632  cres 5633  ccom 5635  Rel wrel 5636  cfv 6493  cmpo 7355  0cc0 11047  1c1 11048  0cn0 12409  seqcseq 13898  𝑟crelexp 14896
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 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5637  df-rel 5638  df-dm 5641  df-oprab 7357  df-mpo 7358  df-relexp 14897
This theorem is referenced by:  relexpsucrd  14910  relexpsucld  14911  relexpreld  14917  relexpdmd  14921  relexprnd  14925  relexpfldd  14927  relexpaddd  14931  dfrtrclrec2  14935  relexpindlem  14940
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