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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 37219 | . 2 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
2 | 1 | relopabiv 5818 | 1 ⊢ Rel ≀ 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∃wex 1782 class class class wbr 5147 Rel wrel 5680 ≀ ccoss 36981 |
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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-in 3954 df-ss 3964 df-opab 5210 df-xp 5681 df-rel 5682 df-coss 37219 |
This theorem is referenced by: relcoels 37232 cocossss 37244 cnvcosseq 37245 refrelcoss3 37271 symrelcoss3 37273 1cosscnvxrn 37283 eleccossin 37291 cosselrels 37304 cnvrefrelcoss2 37345 eqvrelcoss3 37426 |
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