Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
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 35674 | . 2 ⊢ ≀ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)} | |
2 | df-rex 3144 | . . . 4 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))) | |
3 | anandi 674 | . . . . . 6 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦))) | |
4 | brres 5860 | . . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑢(𝑅 ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))) | |
5 | 4 | elv 3499 | . . . . . . 7 ⊢ (𝑢(𝑅 ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥)) |
6 | brres 5860 | . . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑢(𝑅 ↾ 𝐴)𝑦 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦))) | |
7 | 6 | elv 3499 | . . . . . . 7 ⊢ (𝑢(𝑅 ↾ 𝐴)𝑦 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦)) |
8 | 5, 7 | anbi12i 628 | . . . . . 6 ⊢ ((𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑦))) |
9 | 3, 8 | bitr4i 280 | . . . . 5 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) ↔ (𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)) |
10 | 9 | exbii 1848 | . . . 4 ⊢ (∃𝑢(𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) ↔ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)) |
11 | 2, 10 | bitri 277 | . . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)) |
12 | 11 | opabbii 5133 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢(𝑅 ↾ 𝐴)𝑥 ∧ 𝑢(𝑅 ↾ 𝐴)𝑦)} |
13 | 1, 12 | eqtr4i 2847 | 1 ⊢ ≀ (𝑅 ↾ 𝐴) = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝐴 (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 class class class wbr 5066 {copab 5128 ↾ cres 5557 ≀ ccoss 35468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-res 5567 df-coss 35674 |
This theorem is referenced by: dfcoels 35690 |
Copyright terms: Public domain | W3C validator |