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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 14897 | . 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 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 ℕ0cn0 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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