Theorem bdccsb 10651

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 10637	. . . 4 |- BOUNDED z e. A
3	2	bdsbc 10649	. . 3 |- BOUNDED [. y / x ]. z e. A
4	3	bdcab 10640	. 2 |- BOUNDED { z | [. y / x ]. z e. A }
5		df-csb 2909	. 2 |- [_ y / x ]_ A = { z | [. y / x ]. z e. A }
6	4, 5	bdceqir 10635	1 |- BOUNDED [_ y / x ]_ A

Syntax hints:   e. wcel 1433   {cab 2067   [.wsbc 2815   [_csb 2908  BOUNDED wbdc 10631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063  ax-bd0 10604  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-clab 2068  df-cleq 2074  df-clel 2077  df-sbc 2816  df-csb 2909  df-bdc 10632

This theorem is referenced by: (None)