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Theorem bdciun 13061
Description: The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
Hypothesis
Ref Expression
bdciun.1  |- BOUNDED  A
Assertion
Ref Expression
bdciun  |- BOUNDED 
U_ x  e.  y  A
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)

Proof of Theorem bdciun
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 bdciun.1 . . . . 5  |- BOUNDED  A
21bdeli 13029 . . . 4  |- BOUNDED  z  e.  A
32ax-bdex 13002 . . 3  |- BOUNDED  E. x  e.  y  z  e.  A
43bdcab 13032 . 2  |- BOUNDED  { z  |  E. x  e.  y  z  e.  A }
5 df-iun 3810 . 2  |-  U_ x  e.  y  A  =  { z  |  E. x  e.  y  z  e.  A }
64, 5bdceqir 13027 1  |- BOUNDED 
U_ x  e.  y  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1480   {cab 2123   E.wrex 2415   U_ciun 3808  BOUNDED wbdc 13023
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 2119  ax-bd0 12996  ax-bdex 13002  ax-bdsb 13005
This theorem depends on definitions:  df-bi 116  df-clab 2124  df-cleq 2130  df-clel 2133  df-iun 3810  df-bdc 13024
This theorem is referenced by: (None)
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