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Mirrors > Home > MPE Home > Th. List > df-se | Structured version Visualization version GIF version |
Description: Define the set-like predicate. (Contributed by Mario Carneiro, 19-Nov-2014.) |
Ref | Expression |
---|---|
df-se | ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cR | . . 3 class 𝑅 | |
3 | 1, 2 | wse 5543 | . 2 wff 𝑅 Se 𝐴 |
4 | vy | . . . . . . 7 setvar 𝑦 | |
5 | 4 | cv 1541 | . . . . . 6 class 𝑦 |
6 | vx | . . . . . . 7 setvar 𝑥 | |
7 | 6 | cv 1541 | . . . . . 6 class 𝑥 |
8 | 5, 7, 2 | wbr 5079 | . . . . 5 wff 𝑦𝑅𝑥 |
9 | 8, 4, 1 | crab 3070 | . . . 4 class {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} |
10 | cvv 3431 | . . . 4 class V | |
11 | 9, 10 | wcel 2110 | . . 3 wff {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V |
12 | 11, 6, 1 | wral 3066 | . 2 wff ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V |
13 | 3, 12 | wb 205 | 1 wff (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V) |
Colors of variables: wff setvar class |
This definition is referenced by: seex 5552 exse 5553 sess1 5558 sess2 5559 nfse 5565 epse 5573 seinxp 5671 dfse2 6007 exse2 7758 lrrecse 34095 bj-seex 35106 |
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