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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 14967 | . 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 7543 | 1 ⊢ Rel dom ↑𝑟 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 Vcvv 3475 ∪ cun 3947 ifcif 4529 ↦ cmpt 5232 I cid 5574 dom cdm 5677 ran crn 5678 ↾ cres 5679 ∘ ccom 5681 Rel wrel 5682 ‘cfv 6544 ∈ cmpo 7411 0cc0 11110 1c1 11111 ℕ0cn0 12472 seqcseq 13966 ↑𝑟crelexp 14966 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-dm 5687 df-oprab 7413 df-mpo 7414 df-relexp 14967 |
This theorem is referenced by: relexpsucrd 14980 relexpsucld 14981 relexpreld 14987 relexpdmd 14991 relexprnd 14995 relexpfldd 14997 relexpaddd 15001 dfrtrclrec2 15005 relexpindlem 15010 |
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