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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 36294 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥)) | |
2 | 1 | el2v 3416 | . . . . 5 ⊢ (𝑥 ≀ 𝑅𝑦 ↔ 𝑦 ≀ 𝑅𝑥) |
3 | 2 | biimpi 219 | . . . 4 ⊢ (𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) |
4 | 3 | gen2 1804 | . . 3 ⊢ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥) |
5 | cnvsym 5979 | . . 3 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ∀𝑥∀𝑦(𝑥 ≀ 𝑅𝑦 → 𝑦 ≀ 𝑅𝑥)) | |
6 | 4, 5 | mpbir 234 | . 2 ⊢ ◡ ≀ 𝑅 ⊆ ≀ 𝑅 |
7 | relcoss 36283 | . . 3 ⊢ Rel ≀ 𝑅 | |
8 | relcnveq 36194 | . . 3 ⊢ (Rel ≀ 𝑅 → (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ◡ ≀ 𝑅 = ≀ 𝑅)) | |
9 | 7, 8 | ax-mp 5 | . 2 ⊢ (◡ ≀ 𝑅 ⊆ ≀ 𝑅 ↔ ◡ ≀ 𝑅 = ≀ 𝑅) |
10 | 6, 9 | mpbi 233 | 1 ⊢ ◡ ≀ 𝑅 = ≀ 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 = wceq 1543 Vcvv 3408 ⊆ wss 3866 class class class wbr 5053 ◡ccnv 5550 Rel wrel 5556 ≀ ccoss 36070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 df-cnv 5559 df-coss 36274 |
This theorem is referenced by: br2coss 36298 |
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