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Mirrors > Home > MPE Home > Th. List > rabeqc | Structured version Visualization version GIF version |
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 |
---|---|
rabeqc.1 | ⊢ (𝑥 ∈ 𝐴 → 𝜑) |
Ref | Expression |
---|---|
rabeqc | ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3115 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | abeq1 2923 | . . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴 ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 ∈ 𝐴)) | |
3 | rabeqc.1 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝜑) | |
4 | 3 | pm4.71i 563 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
5 | 4 | bicomi 227 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 ∈ 𝐴) |
6 | 2, 5 | mpgbir 1801 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴 |
7 | 1, 6 | eqtri 2821 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {cab 2776 {crab 3110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 |
This theorem is referenced by: 2clwwlk2 28133 numclwwlk3lem2lem 28168 fply1 30982 elnanelprv 32789 |
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