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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1cossres | Structured version Visualization version GIF version |
Description: The class of cosets by a restriction. (Contributed by Peter Mazsa, 20-Apr-2019.) |
Ref | Expression |
---|---|
1cossres | ⊢ ≀ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coss 37794 | . 2 ⊢ ≀ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)} | |
2 | df-rex 3065 | . . . 4 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))) | |
3 | anandi 673 | . . . . . 6 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦))) | |
4 | brres 5982 | . . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑢(𝑅 ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))) | |
5 | 4 | elv 3474 | . . . . . . 7 ⊢ (𝑢(𝑅 ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥)) |
6 | brres 5982 | . . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑢(𝑅 ↾ 𝐴)𝑦 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦))) | |
7 | 6 | elv 3474 | . . . . . . 7 ⊢ (𝑢(𝑅 ↾ 𝐴)𝑦 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦)) |
8 | 5, 7 | anbi12i 626 | . . . . . 6 ⊢ ((𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦))) |
9 | 3, 8 | bitr4i 278 | . . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) ↔ (𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)) |
10 | 9 | exbii 1842 | . . . 4 ⊢ (∃𝑢(𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) ↔ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)) |
11 | 2, 10 | bitri 275 | . . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)) |
12 | 11 | opabbii 5208 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)} |
13 | 1, 12 | eqtr4i 2757 | 1 ⊢ ≀ (𝑅 ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∃wrex 3064 Vcvv 3468 class class class wbr 5141 {copab 5203 ↾ cres 5671 ≀ ccoss 37556 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-res 5681 df-coss 37794 |
This theorem is referenced by: dfcoels 37813 |
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