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 e. A
3	2	bdsbc 13056	. . 3 |- BOUNDED [. y / x ]. z e. A
4	3	bdcab 13047	. 2 |- BOUNDED { z | [. y / x ]. z e. A }
5		df-csb 3004	. 2 |- [_ y / x ]_ A = { z | [. y / x ]. z e. A }
6	4, 5	bdceqir 13042	1 |- BOUNDED [_ y / x ]_ A

Syntax hints:   e. 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)