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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdccsb | GIF version |
Description: A class resulting from proper substitution of a setvar for a setvar in a bounded class is bounded. (Contributed by BJ, 16-Oct-2019.) |
Ref | Expression |
---|---|
bdccsb.1 | ⊢ BOUNDED 𝐴 |
Ref | Expression |
---|---|
bdccsb | ⊢ BOUNDED ⦋𝑦 / 𝑥⦌𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdccsb.1 | . . . . 5 ⊢ BOUNDED 𝐴 | |
2 | 1 | bdeli 15051 | . . . 4 ⊢ BOUNDED 𝑧 ∈ 𝐴 |
3 | 2 | bdsbc 15063 | . . 3 ⊢ BOUNDED [𝑦 / 𝑥]𝑧 ∈ 𝐴 |
4 | 3 | bdcab 15054 | . 2 ⊢ BOUNDED {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴} |
5 | df-csb 3073 | . 2 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴} | |
6 | 4, 5 | bdceqir 15049 | 1 ⊢ BOUNDED ⦋𝑦 / 𝑥⦌𝐴 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 {cab 2175 [wsbc 2977 ⦋csb 3072 BOUNDED wbdc 15045 |
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 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-17 1537 ax-ial 1545 ax-ext 2171 ax-bd0 15018 ax-bdsb 15027 |
This theorem depends on definitions: df-bi 117 df-clab 2176 df-cleq 2182 df-clel 2185 df-sbc 2978 df-csb 3073 df-bdc 15046 |
This theorem is referenced by: (None) |
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