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Mirrors > Home > MPE Home > Th. List > Mathboxes > brcoss3 | Structured version Visualization version GIF version |
Description: Alternate form of the 𝐴 and 𝐵 are cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 26-Mar-2019.) |
Ref | Expression |
---|---|
brcoss3 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ([𝐴]◡𝑅 ∩ [𝐵]◡𝑅) ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcnvg 5893 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ V) → (𝐴◡𝑅𝑢 ↔ 𝑢𝑅𝐴)) | |
2 | 1 | elvd 3484 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡𝑅𝑢 ↔ 𝑢𝑅𝐴)) |
3 | brcnvg 5893 | . . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑢 ∈ V) → (𝐵◡𝑅𝑢 ↔ 𝑢𝑅𝐵)) | |
4 | 3 | elvd 3484 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝐵◡𝑅𝑢 ↔ 𝑢𝑅𝐵)) |
5 | 2, 4 | bi2anan9 638 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴◡𝑅𝑢 ∧ 𝐵◡𝑅𝑢) ↔ (𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) |
6 | 5 | exbidv 1919 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑢(𝐴◡𝑅𝑢 ∧ 𝐵◡𝑅𝑢) ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) |
7 | ecinn0 38335 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴]◡𝑅 ∩ [𝐵]◡𝑅) ≠ ∅ ↔ ∃𝑢(𝐴◡𝑅𝑢 ∧ 𝐵◡𝑅𝑢))) | |
8 | brcoss 38413 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) | |
9 | 6, 7, 8 | 3bitr4rd 312 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ([𝐴]◡𝑅 ∩ [𝐵]◡𝑅) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∩ cin 3962 ∅c0 4339 class class class wbr 5148 ◡ccnv 5688 [cec 8742 ≀ ccoss 38162 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ec 8746 df-coss 38393 |
This theorem is referenced by: br2coss 38420 trcoss2 38466 |
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