Theorem bdccsb 16630

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 16616	. . . 4 ⊢ BOUNDED 𝑧 ∈ 𝐴
3	2	bdsbc 16628	. . 3 ⊢ BOUNDED [𝑦 / 𝑥]𝑧 ∈ 𝐴
4	3	bdcab 16619	. 2 ⊢ BOUNDED {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴}
5		df-csb 3139	. 2 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴}
6	4, 5	bdceqir 16614	1 ⊢ BOUNDED ⦋𝑦 / 𝑥⦌𝐴

Syntax hints:   ∈ wcel 2203  {cab 2218  [wsbc 3042  ⦋csb 3138  BOUNDED wbdc 16610

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2214  ax-bd0 16583  ax-bdsb 16592

This theorem depends on definitions:  df-bi 117  df-clab 2219  df-cleq 2225  df-clel 2228  df-sbc 3043  df-csb 3139  df-bdc 16611

This theorem is referenced by: (None)