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Mirrors > Home > MPE Home > Th. List > Mathboxes > brcosscnvcoss | Structured version Visualization version GIF version |
Description: For sets, the 𝐴 and 𝐵 cosets by 𝑅 binary relation and the 𝐵 and 𝐴 cosets by 𝑅 binary relation are the same. (Contributed by Peter Mazsa, 27-Dec-2018.) |
Ref | Expression |
---|---|
brcosscnvcoss | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ 𝐵 ≀ 𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exancom 1857 | . . 3 ⊢ (∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴)) | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵) ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴))) |
3 | brcoss 38131 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) | |
4 | brcoss 38131 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 ≀ 𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴))) | |
5 | 4 | ancoms 457 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ≀ 𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐵 ∧ 𝑢𝑅𝐴))) |
6 | 2, 3, 5 | 3bitr4d 310 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ 𝐵 ≀ 𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∃wex 1774 ∈ wcel 2099 class class class wbr 5155 ≀ ccoss 37878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pr 5435 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-br 5156 df-opab 5218 df-coss 38111 |
This theorem is referenced by: cocossss 38136 cnvcosseq 38137 rncossdmcoss 38155 symrelcoss3 38165 eleccossin 38183 |
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