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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 35674 | . 2 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
2 | 1 | relopabi 5694 | 1 ⊢ Rel ≀ 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∃wex 1780 class class class wbr 5066 Rel wrel 5560 ≀ ccoss 35468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-xp 5561 df-rel 5562 df-coss 35674 |
This theorem is referenced by: relcoels 35684 cocossss 35696 cnvcosseq 35697 refrelcoss3 35718 symrelcoss3 35720 1cosscnvxrn 35730 eleccossin 35738 cosselrels 35751 cnvrefrelcoss2 35788 eqvrelcoss3 35868 |
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