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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 3073 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | abeq1 2873 | . . 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 1807 | . 2 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴 |
7 | 1, 6 | eqtri 2767 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {cab 2716 {crab 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-rab 3073 |
This theorem is referenced by: 2clwwlk2 28462 numclwwlk3lem2lem 28497 fply1 31412 elnanelprv 33134 bday0s 33792 ipolub0 45996 ipoglb0 45998 |
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