Theorem bdccsb 15516

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 15502	. . . 4 ⊢ BOUNDED 𝑧 ∈ 𝐴
3	2	bdsbc 15514	. . 3 ⊢ BOUNDED [𝑦 / 𝑥]𝑧 ∈ 𝐴
4	3	bdcab 15505	. 2 ⊢ BOUNDED {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴}
5		df-csb 3085	. 2 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴}
6	4, 5	bdceqir 15500	1 ⊢ BOUNDED ⦋𝑦 / 𝑥⦌𝐴

Syntax hints:   ∈ wcel 2167  {cab 2182  [wsbc 2989  ⦋csb 3084  BOUNDED wbdc 15496

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-bd0 15469  ax-bdsb 15478

This theorem depends on definitions:  df-bi 117  df-clab 2183  df-cleq 2189  df-clel 2192  df-sbc 2990  df-csb 3085  df-bdc 15497

This theorem is referenced by: (None)