Theorem bdccsb 13058

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 13044	. . . 4 ⊢ BOUNDED 𝑧 ∈ 𝐴
3	2	bdsbc 13056	. . 3 ⊢ BOUNDED [𝑦 / 𝑥]𝑧 ∈ 𝐴
4	3	bdcab 13047	. 2 ⊢ BOUNDED {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴}
5		df-csb 3004	. 2 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴}
6	4, 5	bdceqir 13042	1 ⊢ BOUNDED ⦋𝑦 / 𝑥⦌𝐴

Syntax hints:   ∈ wcel 1480  {cab 2125  [wsbc 2909  ⦋csb 3003  BOUNDED wbdc 13038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121  ax-bd0 13011  ax-bdsb 13020

This theorem depends on definitions:  df-bi 116  df-clab 2126  df-cleq 2132  df-clel 2135  df-sbc 2910  df-csb 3004  df-bdc 13039

This theorem is referenced by: (None)