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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 34797 | . 2 ⊢ ≀ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)} | |
2 | df-rex 3096 | . . . 4 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))) | |
3 | anandi 666 | . . . . . 6 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦))) | |
4 | brres 5649 | . . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑢(𝑅 ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))) | |
5 | 4 | elv 3402 | . . . . . . 7 ⊢ (𝑢(𝑅 ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥)) |
6 | brres 5649 | . . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑢(𝑅 ↾ 𝐴)𝑦 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦))) | |
7 | 6 | elv 3402 | . . . . . . 7 ⊢ (𝑢(𝑅 ↾ 𝐴)𝑦 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦)) |
8 | 5, 7 | anbi12i 620 | . . . . . 6 ⊢ ((𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦))) |
9 | 3, 8 | bitr4i 270 | . . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) ↔ (𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)) |
10 | 9 | exbii 1892 | . . . 4 ⊢ (∃𝑢(𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) ↔ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)) |
11 | 2, 10 | bitri 267 | . . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)) |
12 | 11 | opabbii 4953 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)} |
13 | 1, 12 | eqtr4i 2805 | 1 ⊢ ≀ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1601 ∃wex 1823 ∈ wcel 2107 ∃wrex 3091 Vcvv 3398 class class class wbr 4886 {copab 4948 ↾ cres 5357 ≀ ccoss 34606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4887 df-opab 4949 df-xp 5361 df-res 5367 df-coss 34797 |
This theorem is referenced by: dfcoels 34813 |
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