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Theorem iunconst 4065
Description: Indexed union of a constant class, i.e. where  B does not depend on  x. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iunconst  |-  ( A  =/=  (/)  ->  U_ x  e.  A  B  =  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem iunconst
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.9rzv 3686 . . 3  |-  ( A  =/=  (/)  ->  ( y  e.  B  <->  E. x  e.  A  y  e.  B )
)
2 eliun 4061 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
31, 2syl6rbbr 256 . 2  |-  ( A  =/=  (/)  ->  ( y  e.  U_ x  e.  A  B 
<->  y  e.  B ) )
43eqrdv 2406 1  |-  ( A  =/=  (/)  ->  U_ x  e.  A  B  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    =/= wne 2571   E.wrex 2671   (/)c0 3592   U_ciun 4057
This theorem is referenced by:  iununi  4139  abianfplem  6678  oe1m  6751  oarec  6768  oelim2  6801  mblfinlem  26147  bnj1143  28871
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-v 2922  df-dif 3287  df-nul 3593  df-iun 4059
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