Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
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 5066 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑢𝑅𝑥 ↔ 𝑢𝑅𝐴)) | |
2 | breq2 5066 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑢𝑅𝑦 ↔ 𝑢𝑅𝐵)) | |
3 | 1, 2 | bi2anan9 639 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ (𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) |
4 | 3 | exbidv 1929 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐵))) |
5 | df-coss 36287 | . 2 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
6 | 4, 5 | brabga 5424 | 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 1543 ∃wex 1787 ∈ wcel 2111 class class class wbr 5062 ≀ ccoss 36083 |
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 2113 ax-9 2121 ax-ext 2709 ax-sep 5201 ax-nul 5208 ax-pr 5331 |
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 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3071 df-v 3417 df-dif 3878 df-un 3880 df-nul 4247 df-if 4449 df-sn 4551 df-pr 4553 df-op 4557 df-br 5063 df-opab 5125 df-coss 36287 |
This theorem is referenced by: brcoss2 36305 brcoss3 36306 brcosscnvcoss 36307 cocossss 36309 br1cossres 36312 eldmcoss2 36327 brcosscnv 36340 trcoss 36350 |
Copyright terms: Public domain | W3C validator |