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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvcosseq | Structured version Visualization version GIF version |
Description: The converse of cosets by 𝑅 are cosets by 𝑅. (Contributed by Peter Mazsa, 3-May-2019.) |
Ref | Expression |
---|---|
cnvcosseq | ⊢ ◡ ≀ 𝑅 = ≀ 𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcosscnvcoss 38339 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥)) | |
2 | 1 | el2v 3490 | . . . . 5 ⊢ (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥) |
3 | 2 | biimpi 216 | . . . 4 ⊢ (𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) |
4 | 3 | gen2 1794 | . . 3 ⊢ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) |
5 | cnvsym 6143 | . . 3 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥)) | |
6 | 4, 5 | mpbir 231 | . 2 ⊢ ◡ ≀ 𝑅 ⊆ ≀ 𝑅 |
7 | relcoss 38328 | . . 3 ⊢ Rel ≀ 𝑅 | |
8 | relcnveq 38227 | . . 3 ⊢ (Rel ≀ 𝑅 → (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ◡ ≀ 𝑅 = ≀ 𝑅)) | |
9 | 7, 8 | ax-mp 5 | . 2 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ◡ ≀ 𝑅 = ≀ 𝑅) |
10 | 6, 9 | mpbi 230 | 1 ⊢ ◡ ≀ 𝑅 = ≀ 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 = wceq 1537 Vcvv 3482 ⊆ wss 3970 class class class wbr 5169 ◡ccnv 5698 Rel wrel 5704 ≀ ccoss 38084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5170 df-opab 5232 df-xp 5705 df-rel 5706 df-cnv 5707 df-coss 38316 |
This theorem is referenced by: br2coss 38343 |
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