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Mirrors > Home > MPE Home > Th. List > Mathboxes > brcoss | Structured version Visualization version GIF version |
Description: 𝐴 and 𝐵 are cosets by 𝑅: a binary relation. (Contributed by Peter Mazsa, 27-Dec-2018.) |
Ref | Expression |
---|---|
brcoss | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5153 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑢𝑅𝑥 ↔ 𝑢𝑅𝐴)) | |
2 | breq2 5153 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑢𝑅𝑦 ↔ 𝑢𝑅𝐵)) | |
3 | 1, 2 | bi2anan9 638 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ (𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) |
4 | 3 | exbidv 1925 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) |
5 | df-coss 37281 | . 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
6 | 4, 5 | brabga 5535 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 class class class wbr 5149 ≀ ccoss 37043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-coss 37281 |
This theorem is referenced by: brcoss2 37302 brcoss3 37303 brcosscnvcoss 37304 cocossss 37306 br1cossres 37309 eldmcoss2 37329 brcosscnv 37342 trcoss 37352 |
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