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Theorem bdciun 15776
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 15744 . . . 4  |- BOUNDED  z  e.  A
32ax-bdex 15717 . . 3  |- BOUNDED  E. x  e.  y  z  e.  A
43bdcab 15747 . 2  |- BOUNDED  { z  |  E. x  e.  y  z  e.  A }
5 df-iun 3928 . 2  |-  U_ x  e.  y  A  =  { z  |  E. x  e.  y  z  e.  A }
64, 5bdceqir 15742 1  |- BOUNDED 
U_ x  e.  y  A
Colors of variables: wff set class
Syntax hints:    e. wcel 2175   {cab 2190   E.wrex 2484   U_ciun 3926  BOUNDED wbdc 15738
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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186  ax-bd0 15711  ax-bdex 15717  ax-bdsb 15720
This theorem depends on definitions:  df-bi 117  df-clab 2191  df-cleq 2197  df-clel 2200  df-iun 3928  df-bdc 15739
This theorem is referenced by: (None)
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