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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 36272 | . 2 ⊢ ≀ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)} | |
2 | df-rex 3064 | . . . 4 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))) | |
3 | anandi 676 | . . . . . 6 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦))) | |
4 | brres 5855 | . . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑢(𝑅 ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))) | |
5 | 4 | elv 3411 | . . . . . . 7 ⊢ (𝑢(𝑅 ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥)) |
6 | brres 5855 | . . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑢(𝑅 ↾ 𝐴)𝑦 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦))) | |
7 | 6 | elv 3411 | . . . . . . 7 ⊢ (𝑢(𝑅 ↾ 𝐴)𝑦 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦)) |
8 | 5, 7 | anbi12i 630 | . . . . . 6 ⊢ ((𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦))) |
9 | 3, 8 | bitr4i 281 | . . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) ↔ (𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)) |
10 | 9 | exbii 1855 | . . . 4 ⊢ (∃𝑢(𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) ↔ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)) |
11 | 2, 10 | bitri 278 | . . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)) |
12 | 11 | opabbii 5117 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)} |
13 | 1, 12 | eqtr4i 2768 | 1 ⊢ ≀ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2110 ∃wrex 3059 Vcvv 3405 class class class wbr 5050 {copab 5112 ↾ cres 5550 ≀ ccoss 36068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pr 5319 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3063 df-rex 3064 df-rab 3067 df-v 3407 df-dif 3866 df-un 3868 df-in 3870 df-nul 4235 df-if 4437 df-sn 4539 df-pr 4541 df-op 4545 df-br 5051 df-opab 5113 df-xp 5554 df-res 5560 df-coss 36272 |
This theorem is referenced by: dfcoels 36288 |
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