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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-bj-gab | Structured version Visualization version GIF version |
Description: Definition of generalized class abstractions: typically, 𝑥 is a bound variable in 𝐴 and 𝜑 and {𝐴 ∣ 𝑥 ∣ 𝜑} denotes "the class of 𝐴(𝑥)'s such that 𝜑(𝑥)". (Contributed by BJ, 4-Oct-2024.) |
Ref | Expression |
---|---|
df-bj-gab | ⊢ {𝐴 ∣ 𝑥 ∣ 𝜑} = {𝑦 ∣ ∃𝑥(𝐴 = 𝑦 ∧ 𝜑)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cA | . . 3 class 𝐴 | |
4 | 1, 2, 3 | bj-cgab 35129 | . 2 class {𝐴 ∣ 𝑥 ∣ 𝜑} |
5 | vy | . . . . . . 7 setvar 𝑦 | |
6 | 5 | cv 1538 | . . . . . 6 class 𝑦 |
7 | 3, 6 | wceq 1539 | . . . . 5 wff 𝐴 = 𝑦 |
8 | 7, 1 | wa 396 | . . . 4 wff (𝐴 = 𝑦 ∧ 𝜑) |
9 | 8, 2 | wex 1782 | . . 3 wff ∃𝑥(𝐴 = 𝑦 ∧ 𝜑) |
10 | 9, 5 | cab 2715 | . 2 class {𝑦 ∣ ∃𝑥(𝐴 = 𝑦 ∧ 𝜑)} |
11 | 4, 10 | wceq 1539 | 1 wff {𝐴 ∣ 𝑥 ∣ 𝜑} = {𝑦 ∣ ∃𝑥(𝐴 = 𝑦 ∧ 𝜑)} |
Colors of variables: wff setvar class |
This definition is referenced by: bj-gabss 35131 bj-gabeqis 35134 bj-elgab 35135 |
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