Theorem bdccsb 9980

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.)

Hypothesis

Ref	Expression
bdccsb.1	⊢ BOUNDED 𝐴

Assertion

Ref	Expression
bdccsb	⊢ BOUNDED ⦋𝑦 / 𝑥⦌𝐴

Proof of Theorem bdccsb

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdccsb.1	. . . . 5 ⊢ BOUNDED 𝐴
2	1	bdeli 9966	. . . 4 ⊢ BOUNDED 𝑧 ∈ 𝐴
3	2	bdsbc 9978	. . 3 ⊢ BOUNDED [𝑦 / 𝑥]𝑧 ∈ 𝐴
4	3	bdcab 9969	. 2 ⊢ BOUNDED {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴}
5		df-csb 2853	. 2 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴}
6	4, 5	bdceqir 9964	1 ⊢ BOUNDED ⦋𝑦 / 𝑥⦌𝐴

Syntax hints:   ∈ wcel 1393  {cab 2026  [wsbc 2764  ⦋csb 2852  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-bd0 9933  ax-bdsb 9942

This theorem depends on definitions:  df-bi 110  df-clab 2027  df-cleq 2033  df-clel 2036  df-sbc 2765  df-csb 2853  df-bdc 9961

This theorem is referenced by: (None)