![]() |
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 34680 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑅𝐵 ↔ ∃𝑥(𝑥◡𝑅𝐴 ∧ 𝑥◡𝑅𝐵))) | |
2 | brcnvg 5505 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥◡𝑅𝐴 ↔ 𝐴𝑅𝑥)) | |
3 | 2 | el2v1 34492 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥◡𝑅𝐴 ↔ 𝐴𝑅𝑥)) |
4 | brcnvg 5505 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐵 ∈ 𝑊) → (𝑥◡𝑅𝐵 ↔ 𝐵𝑅𝑥)) | |
5 | 4 | el2v1 34492 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → (𝑥◡𝑅𝐵 ↔ 𝐵𝑅𝑥)) |
6 | 3, 5 | bi2anan9 630 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝑥◡𝑅𝐴 ∧ 𝑥◡𝑅𝐵) ↔ (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
7 | 6 | exbidv 2017 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥(𝑥◡𝑅𝐴 ∧ 𝑥◡𝑅𝐵) ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
8 | 1, 7 | bitrd 271 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∃wex 1875 ∈ wcel 2157 Vcvv 3385 class class class wbr 4843 ◡ccnv 5311 ≀ ccoss 34469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-cnv 5320 df-coss 34663 |
This theorem is referenced by: brcosscnv2 34717 br1cosscnvxrn 34718 |
Copyright terms: Public domain | W3C validator |