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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 5034 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑢𝑅𝑥 ↔ 𝑢𝑅𝐴)) | |
2 | breq2 5034 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑢𝑅𝑦 ↔ 𝑢𝑅𝐵)) | |
3 | 1, 2 | bi2anan9 638 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ (𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) |
4 | 3 | exbidv 1922 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) |
5 | df-coss 35819 | . 2 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
6 | 4, 5 | brabga 5386 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ 𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 class class class wbr 5030 ≀ ccoss 35613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-coss 35819 |
This theorem is referenced by: brcoss2 35837 brcoss3 35838 brcosscnvcoss 35839 cocossss 35841 br1cossres 35844 eldmcoss2 35859 brcosscnv 35872 trcoss 35882 |
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