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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cosscnv | Structured version Visualization version GIF version | ||
| Description: Class of cosets by the converse of 𝑅 (Contributed by Peter Mazsa, 17-Jun-2020.) |
| Ref | Expression |
|---|---|
| cosscnv | ⊢ ≀ ◡𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-coss 38412 | . 2 ⊢ ≀ ◡𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦)} | |
| 2 | brcnvg 5890 | . . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑢◡𝑅𝑥 ↔ 𝑥𝑅𝑢)) | |
| 3 | 2 | el2v 3487 | . . . . 5 ⊢ (𝑢◡𝑅𝑥 ↔ 𝑥𝑅𝑢) |
| 4 | brcnvg 5890 | . . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑦 ∈ V) → (𝑢◡𝑅𝑦 ↔ 𝑦𝑅𝑢)) | |
| 5 | 4 | el2v 3487 | . . . . 5 ⊢ (𝑢◡𝑅𝑦 ↔ 𝑦𝑅𝑢) |
| 6 | 3, 5 | anbi12i 628 | . . . 4 ⊢ ((𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦) ↔ (𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)) |
| 7 | 6 | exbii 1848 | . . 3 ⊢ (∃𝑢(𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦) ↔ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)) |
| 8 | 7 | opabbii 5210 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)} |
| 9 | 1, 8 | eqtri 2765 | 1 ⊢ ≀ ◡𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)} |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 Vcvv 3480 class class class wbr 5143 {copab 5205 ◡ccnv 5684 ≀ ccoss 38182 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-cnv 5693 df-coss 38412 |
| This theorem is referenced by: (None) |
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