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Mirrors > Home > MPE Home > Th. List > Mathboxes > relcoss | Structured version Visualization version GIF version |
Description: Cosets by 𝑅 is a relation. (Contributed by Peter Mazsa, 27-Dec-2018.) |
Ref | Expression |
---|---|
relcoss | ⊢ Rel ≀ 𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coss 36149 | . 2 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
2 | 1 | relopabiv 5658 | 1 ⊢ Rel ≀ 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∃wex 1786 class class class wbr 5027 Rel wrel 5524 ≀ ccoss 35945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3399 df-in 3848 df-ss 3858 df-opab 5090 df-xp 5525 df-rel 5526 df-coss 36149 |
This theorem is referenced by: relcoels 36159 cocossss 36171 cnvcosseq 36172 refrelcoss3 36193 symrelcoss3 36195 1cosscnvxrn 36205 eleccossin 36213 cosselrels 36226 cnvrefrelcoss2 36263 eqvrelcoss3 36343 |
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