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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdciun | GIF version |
Description: The indexed union 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 |
---|---|
bdciun | ⊢ BOUNDED ∪ 𝑥 ∈ 𝑦 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdciun.1 | . . . . 5 ⊢ BOUNDED 𝐴 | |
2 | 1 | bdeli 14220 | . . . 4 ⊢ BOUNDED 𝑧 ∈ 𝐴 |
3 | 2 | ax-bdex 14193 | . . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝐴 |
4 | 3 | bdcab 14223 | . 2 ⊢ BOUNDED {𝑧 ∣ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝐴} |
5 | df-iun 3886 | . 2 ⊢ ∪ 𝑥 ∈ 𝑦 𝐴 = {𝑧 ∣ ∃𝑥 ∈ 𝑦 𝑧 ∈ 𝐴} | |
6 | 4, 5 | bdceqir 14218 | 1 ⊢ BOUNDED ∪ 𝑥 ∈ 𝑦 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 {cab 2163 ∃wrex 2456 ∪ ciun 3884 BOUNDED wbdc 14214 |
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 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-17 1526 ax-ial 1534 ax-ext 2159 ax-bd0 14187 ax-bdex 14193 ax-bdsb 14196 |
This theorem depends on definitions: df-bi 117 df-clab 2164 df-cleq 2170 df-clel 2173 df-iun 3886 df-bdc 14215 |
This theorem is referenced by: (None) |
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