Theorem bdccsb 12750

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 12736	. . . 4 |- BOUNDED z e. A
3	2	bdsbc 12748	. . 3 |- BOUNDED [. y / x ]. z e. A
4	3	bdcab 12739	. 2 |- BOUNDED { z | [. y / x ]. z e. A }
5		df-csb 2972	. 2 |- [_ y / x ]_ A = { z | [. y / x ]. z e. A }
6	4, 5	bdceqir 12734	1 |- BOUNDED [_ y / x ]_ A

Syntax hints:   e. wcel 1463   {cab 2101   [.wsbc 2878   [_csb 2971  BOUNDED wbdc 12730

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-17 1489  ax-ial 1497  ax-ext 2097  ax-bd0 12703  ax-bdsb 12712

This theorem depends on definitions:  df-bi 116  df-clab 2102  df-cleq 2108  df-clel 2111  df-sbc 2879  df-csb 2972  df-bdc 12731

This theorem is referenced by: (None)