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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 38412 | . 2 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
| 2 | 1 | relopabiv 5830 | 1 ⊢ Rel ≀ 𝑅 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∃wex 1779 class class class wbr 5143 Rel wrel 5690 ≀ ccoss 38182 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-opab 5206 df-xp 5691 df-rel 5692 df-coss 38412 | 
| This theorem is referenced by: relcoels 38425 cocossss 38437 cnvcosseq 38438 refrelcoss3 38464 symrelcoss3 38466 1cosscnvxrn 38476 eleccossin 38484 cosselrels 38497 cnvrefrelcoss2 38538 eqvrelcoss3 38619 | 
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