Theorem bdccsb 14697

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 14683	. . . 4 |- BOUNDED z e. A
3	2	bdsbc 14695	. . 3 |- BOUNDED [. y / x ]. z e. A
4	3	bdcab 14686	. 2 |- BOUNDED { z | [. y / x ]. z e. A }
5		df-csb 3060	. 2 |- [_ y / x ]_ A = { z | [. y / x ]. z e. A }
6	4, 5	bdceqir 14681	1 |- BOUNDED [_ y / x ]_ A

Syntax hints:   e. wcel 2148   {cab 2163   [.wsbc 2964   [_csb 3059  BOUNDED wbdc 14677

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159  ax-bd0 14650  ax-bdsb 14659

This theorem depends on definitions:  df-bi 117  df-clab 2164  df-cleq 2170  df-clel 2173  df-sbc 2965  df-csb 3060  df-bdc 14678

This theorem is referenced by: (None)