![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > brcosscnv | Structured version Visualization version GIF version |
Description: 𝐴 and 𝐵 are cosets by converse 𝑅: a binary relation. (Contributed by Peter Mazsa, 23-Jan-2019.) |
Ref | Expression |
---|---|
brcosscnv | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcoss 37943 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑅𝐵 ↔ ∃𝑥(𝑥◡𝑅𝐴 ∧ 𝑥◡𝑅𝐵))) | |
2 | brcnvg 5886 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥◡𝑅𝐴 ↔ 𝐴𝑅𝑥)) | |
3 | 2 | el2v1 37731 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥◡𝑅𝐴 ↔ 𝐴𝑅𝑥)) |
4 | brcnvg 5886 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐵 ∈ 𝑊) → (𝑥◡𝑅𝐵 ↔ 𝐵𝑅𝑥)) | |
5 | 4 | el2v1 37731 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝑥◡𝑅𝐵 ↔ 𝐵𝑅𝑥)) |
6 | 3, 5 | bi2anan9 636 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑥◡𝑅𝐴 ∧ 𝑥◡𝑅𝐵) ↔ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
7 | 6 | exbidv 1916 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥(𝑥◡𝑅𝐴 ∧ 𝑥◡𝑅𝐵) ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
8 | 1, 7 | bitrd 278 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∃wex 1773 ∈ wcel 2098 Vcvv 3473 class class class wbr 5152 ◡ccnv 5681 ≀ ccoss 37689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-cnv 5690 df-coss 37923 |
This theorem is referenced by: brcosscnv2 37985 br1cosscnvxrn 37986 |
Copyright terms: Public domain | W3C validator |