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Mirrors > Home > MPE Home > Th. List > reldmrelexp | Structured version Visualization version GIF version |
Description: The domain of the repeated composition of a relation is a relation. (Contributed by AV, 12-Jul-2024.) |
Ref | Expression |
---|---|
reldmrelexp | ⊢ Rel dom ↑𝑟 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-relexp 14969 | . 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 7545 | 1 ⊢ Rel dom ↑𝑟 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 Vcvv 3474 ∪ cun 3946 ifcif 4528 ↦ cmpt 5231 I cid 5573 dom cdm 5676 ran crn 5677 ↾ cres 5678 ∘ ccom 5680 Rel wrel 5681 ‘cfv 6543 ∈ cmpo 7413 0cc0 11112 1c1 11113 ℕ0cn0 12474 seqcseq 13968 ↑𝑟crelexp 14968 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-dm 5686 df-oprab 7415 df-mpo 7416 df-relexp 14969 |
This theorem is referenced by: relexpsucrd 14982 relexpsucld 14983 relexpreld 14989 relexpdmd 14993 relexprnd 14997 relexpfldd 14999 relexpaddd 15003 dfrtrclrec2 15007 relexpindlem 15012 |
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