Theorem bdccsb 9980

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 9966	. . . 4 |- BOUNDED z e. A
3	2	bdsbc 9978	. . 3 |- BOUNDED [. y / x ]. z e. A
4	3	bdcab 9969	. 2 |- BOUNDED { z | [. y / x ]. z e. A }
5		df-csb 2853	. 2 |- [_ y / x ]_ A = { z | [. y / x ]. z e. A }
6	4, 5	bdceqir 9964	1 |- BOUNDED [_ y / x ]_ A

Syntax hints:   e. wcel 1393   {cab 2026   [.wsbc 2764   [_csb 2852  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-bd0 9933  ax-bdsb 9942

This theorem depends on definitions:  df-bi 110  df-clab 2027  df-cleq 2033  df-clel 2036  df-sbc 2765  df-csb 2853  df-bdc 9961

This theorem is referenced by: (None)