Theorem reldmrelexp 15186

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 15185	. 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 7561	1 ⊢ Rel dom ↑𝑟

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3453   ∪ cun 4151  ifcif 4484   ↦ cmpt 5188   I cid 5548  dom cdm 5654  ran crn 5655   ↾ cres 5656   ∘ ccom 5658  Rel wrel 5659  ‘cfv 6546   ∈ cmpo 7429  0cc0 11210  1c1 11211  ℕ₀cn0 12616  seqcseq 14154  ↑_𝑟crelexp 15184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-rel 5661  df-dm 5664  df-oprab 7431  df-mpo 7432  df-relexp 15185

This theorem is used by:  relexpsucrd  15198  relexpsucld  15199  relexpreld  15205  relexpdmd  15209  relexprnd  15213  relexpfldd  15215  relexpaddd  15219  dfrtrclrec2  15223  relexpindlem  15228