Theorem bdccsb 16800

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 16786	. . . 4 ⊢ BOUNDED 𝑧 ∈ 𝐴
3	2	bdsbc 16798	. . 3 ⊢ BOUNDED [𝑦 / 𝑥]𝑧 ∈ 𝐴
4	3	bdcab 16789	. 2 ⊢ BOUNDED {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴}
5		df-csb 3148	. 2 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴}
6	4, 5	bdceqir 16784	1 ⊢ BOUNDED ⦋𝑦 / 𝑥⦌𝐴

Syntax hints:   ∈ wcel 2209  {cab 2224  [wsbc 3051  ⦋csb 3147  BOUNDED wbdc 16780

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-bd0 16753  ax-bdsb 16762

This theorem depends on definitions:  df-bi 117  df-clab 2225  df-cleq 2231  df-clel 2234  df-sbc 3052  df-csb 3148  df-bdc 16781

This theorem is referenced by: (None)