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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 35180 | . 2 ⊢ (𝐴 ∈ 𝑉 → ◡𝐴 ∈ Rels ) | |
2 | cosselrels 35181 | . 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 2050 ◡ccnv 5400 ≀ ccoss 34897 Rels crels 34899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-coss 35104 df-rels 35170 |
This theorem is referenced by: dfdisjs2 35387 eldisjs2 35401 |
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