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| Mirrors > Home > ILE Home > Th. List > Mathboxes > bdciin | GIF version | ||
| Description: The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.) |
| Ref | Expression |
|---|---|
| bdciun.1 | ⊢ BOUNDED 𝐴 |
| Ref | Expression |
|---|---|
| bdciin | ⊢ BOUNDED ∩ 𝑥 ∈ 𝑦 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bdciun.1 | . . . . 5 ⊢ BOUNDED 𝐴 | |
| 2 | 1 | bdeli 15715 | . . . 4 ⊢ BOUNDED 𝑧 ∈ 𝐴 |
| 3 | 2 | ax-bdal 15687 | . . 3 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 𝑧 ∈ 𝐴 |
| 4 | 3 | bdcab 15718 | . 2 ⊢ BOUNDED {𝑧 ∣ ∀𝑥 ∈ 𝑦 𝑧 ∈ 𝐴} |
| 5 | df-iin 3929 | . 2 ⊢ ∩ 𝑥 ∈ 𝑦 𝐴 = {𝑧 ∣ ∀𝑥 ∈ 𝑦 𝑧 ∈ 𝐴} | |
| 6 | 4, 5 | bdceqir 15713 | 1 ⊢ BOUNDED ∩ 𝑥 ∈ 𝑦 𝐴 |
| Colors of variables: wff set class |
| Syntax hints: ∈ wcel 2175 {cab 2190 ∀wral 2483 ∩ ciin 3927 BOUNDED wbdc 15709 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1469 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-4 1532 ax-17 1548 ax-ial 1556 ax-ext 2186 ax-bd0 15682 ax-bdal 15687 ax-bdsb 15691 |
| This theorem depends on definitions: df-bi 117 df-clab 2191 df-cleq 2197 df-clel 2200 df-iin 3929 df-bdc 15710 |
| This theorem is referenced by: (None) |
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