Theorem bdccsb 16886

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 16872	. . . 4 |- BOUNDED z e. A
3	2	bdsbc 16884	. . 3 |- BOUNDED [. y / x ]. z e. A
4	3	bdcab 16875	. 2 |- BOUNDED { z | [. y / x ]. z e. A }
5		df-csb 3148	. 2 |- [_ y / x ]_ A = { z | [. y / x ]. z e. A }
6	4, 5	bdceqir 16870	1 |- BOUNDED [_ y / x ]_ A

This proof depends on syntax axioms:   e. wcel 2209   {cab 2224   [.wsbc 3051   [_csb 3147  BOUNDED wbdc 16866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-bd0 16839  ax-bdsb 16848

This proof depends on definitions:  df-bi 117  df-clab 2225  df-cleq 2231  df-clel 2234  df-sbc 3052  df-csb 3148  df-bdc 16867

This theorem is used by: (None)