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| Mirrors > Home > MPE Home > Th. List > Mathboxes > coss1cnvres | Structured version Visualization version GIF version | ||
| Description: Class of cosets by the converse of a restriction. (Contributed by Peter Mazsa, 8-Jun-2020.) |
| Ref | Expression |
|---|---|
| coss1cnvres | ⊢ ≀ ◡(𝑅 ↾ 𝐴) = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-coss 38412 | . 2 ⊢ ≀ ◡(𝑅 ↾ 𝐴) = {〈𝑢, 𝑣〉 ∣ ∃𝑥(𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣)} | |
| 2 | br1cnvres 38270 | . . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥◡(𝑅 ↾ 𝐴)𝑢 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))) | |
| 3 | 2 | elv 3485 | . . . . . . 7 ⊢ (𝑥◡(𝑅 ↾ 𝐴)𝑢 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥)) |
| 4 | br1cnvres 38270 | . . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥◡(𝑅 ↾ 𝐴)𝑣 ↔ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥))) | |
| 5 | 4 | elv 3485 | . . . . . . 7 ⊢ (𝑥◡(𝑅 ↾ 𝐴)𝑣 ↔ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥)) |
| 6 | 3, 5 | anbi12i 628 | . . . . . 6 ⊢ ((𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥))) |
| 7 | an4 656 | . . . . . 6 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥))) | |
| 8 | 6, 7 | bitr4i 278 | . . . . 5 ⊢ ((𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))) |
| 9 | 8 | exbii 1848 | . . . 4 ⊢ (∃𝑥(𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ∃𝑥((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))) |
| 10 | 19.42v 1953 | . . . 4 ⊢ (∃𝑥((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))) | |
| 11 | 9, 10 | bitri 275 | . . 3 ⊢ (∃𝑥(𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))) |
| 12 | 11 | opabbii 5210 | . 2 ⊢ {〈𝑢, 𝑣〉 ∣ ∃𝑥(𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣)} = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))} |
| 13 | 1, 12 | eqtri 2765 | 1 ⊢ ≀ ◡(𝑅 ↾ 𝐴) = {〈𝑢, 𝑣〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))} |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 {copab 5205 ◡ccnv 5684 ↾ cres 5687 ≀ 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 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-xp 5691 df-rel 5692 df-cnv 5693 df-res 5697 df-coss 38412 |
| This theorem is referenced by: coss2cnvepres 38419 |
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