Theorem reldmrelexp 15061

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 15060	. 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 7568	1 ⊢ Rel dom ↑𝑟

Syntax hints:   = wceq 1539  Vcvv 3479   ∪ cun 3948  ifcif 4524   ↦ cmpt 5224   I cid 5576  dom cdm 5684  ran crn 5685   ↾ cres 5686   ∘ ccom 5688  Rel wrel 5689  ‘cfv 6560   ∈ cmpo 7434  0cc0 11156  1c1 11157  ℕ₀cn0 12528  seqcseq 14043  ↑_𝑟crelexp 15059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-dm 5694  df-oprab 7436  df-mpo 7437  df-relexp 15060

This theorem is referenced by:  relexpsucrd  15073  relexpsucld  15074  relexpreld  15080  relexpdmd  15084  relexprnd  15088  relexpfldd  15090  relexpaddd  15094  dfrtrclrec2  15098  relexpindlem  15103