Description: A restricted class abstraction equals the restricting class if its condition follows from the membership of the free setvar variable in the restricting class. (Contributed by AV, 20-Apr-2022)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | rabeqc.1 | ⊢ ( 𝑥 ∈ 𝐴 → 𝜑 ) | |
| Assertion | rabeqc | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rabeqc.1 | ⊢ ( 𝑥 ∈ 𝐴 → 𝜑 ) | |
| 2 | df-rab | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } | |
| 3 | abeq1 | ⊢ ( { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } = 𝐴 ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ 𝑥 ∈ 𝐴 ) ) | |
| 4 | 1 | pm4.71i | ⊢ ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) |
| 5 | 4 | bicomi | ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ 𝑥 ∈ 𝐴 ) |
| 6 | 3 5 | mpgbir | ⊢ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } = 𝐴 |
| 7 | 2 6 | eqtri | ⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = 𝐴 |