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Mirrors > Home > MPE Home > Th. List > Mathboxes > cosscnvelrels | Structured version Visualization version GIF version |
Description: Cosets of converse sets are elements of the relations class. (Contributed by Peter Mazsa, 31-Aug-2021.) |
Ref | Expression |
---|---|
cosscnvelrels | ⊢ (𝐴 ∈ 𝑉 → ≀ ◡𝐴 ∈ Rels ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvelrels 36600 | . 2 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ Rels ) | |
2 | cosselrels 36601 | . 2 ⊢ (◡𝐴 ∈ Rels → ≀ ◡𝐴 ∈ Rels ) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → ≀ ◡𝐴 ∈ Rels ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ◡ccnv 5585 ≀ ccoss 36320 Rels crels 36322 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3433 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5076 df-opab 5138 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-coss 36524 df-rels 36590 |
This theorem is referenced by: dfdisjs2 36807 eldisjs2 36821 |
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