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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 14412 | . 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 7273 | 1 ⊢ Rel dom ↑𝑟 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 Vcvv 3407 ∪ cun 3852 ifcif 4413 ↦ cmpt 5105 I cid 5422 dom cdm 5517 ran crn 5518 ↾ cres 5519 ∘ ccom 5521 Rel wrel 5522 ‘cfv 6328 ∈ cmpo 7145 0cc0 10560 1c1 10561 ℕ0cn0 11919 seqcseq 13403 ↑𝑟crelexp 14411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-v 3409 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 df-br 5026 df-opab 5088 df-xp 5523 df-rel 5524 df-dm 5527 df-oprab 7147 df-mpo 7148 df-relexp 14412 |
This theorem is referenced by: relexpsucrd 14425 relexpsucld 14426 relexpreld 14432 relexpdmd 14436 relexprnd 14440 relexpfldd 14442 relexpaddd 14446 dfrtrclrec2 14450 relexpindlem 14455 |
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