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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdcrab | GIF version |
Description: A class defined by restricted abstraction from a bounded class and a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.) |
Ref | Expression |
---|---|
bdcrab.1 | ⊢ BOUNDED 𝐴 |
bdcrab.2 | ⊢ BOUNDED 𝜑 |
Ref | Expression |
---|---|
bdcrab | ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdcrab.1 | . . . . 5 ⊢ BOUNDED 𝐴 | |
2 | 1 | bdeli 12723 | . . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴 |
3 | bdcrab.2 | . . . 4 ⊢ BOUNDED 𝜑 | |
4 | 2, 3 | ax-bdan 12692 | . . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∧ 𝜑) |
5 | 4 | bdcab 12726 | . 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
6 | df-rab 2397 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
7 | 5, 6 | bdceqir 12721 | 1 ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∈ wcel 1461 {cab 2099 {crab 2392 BOUNDED wbd 12689 BOUNDED wbdc 12717 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1404 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-4 1468 ax-17 1487 ax-ial 1495 ax-ext 2095 ax-bd0 12690 ax-bdan 12692 ax-bdsb 12699 |
This theorem depends on definitions: df-bi 116 df-clab 2100 df-cleq 2106 df-clel 2109 df-rab 2397 df-bdc 12718 |
This theorem is referenced by: bdrabexg 12783 |
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