Theorem reldmrelexp 15084

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 15083	. 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 7553	1 ⊢ Rel dom ↑𝑟

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3457   ∪ cun 3904  ifcif 4489   ↦ cmpt 5194   I cid 5557  dom cdm 5663  ran crn 5664   ↾ cres 5665   ∘ ccom 5667  Rel wrel 5668  ‘cfv 6540   ∈ cmpo 7421  0cc0 11117  1c1 11118  ℕ₀cn0 12521  seqcseq 14057  ↑_𝑟crelexp 15082

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-dm 5673  df-oprab 7423  df-mpo 7424  df-relexp 15083

This theorem is used by:  relexpsucrd  15096  relexpsucld  15097  relexpreld  15103  relexpdmd  15107  relexprnd  15111  relexpfldd  15113  relexpaddd  15117  dfrtrclrec2  15121  relexpindlem  15126