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Theorem iunconst 3992
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 3624 . . 3  |-  ( A  =/=  (/)  ->  ( y  e.  B  <->  E. x  e.  A  y  e.  B )
)
2 eliun 3988 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
31, 2syl6rbbr 255 . 2  |-  ( A  =/=  (/)  ->  ( y  e.  U_ x  e.  A  B 
<->  y  e.  B ) )
43eqrdv 2356 1  |-  ( A  =/=  (/)  ->  U_ x  e.  A  B  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620   (/)c0 3531   U_ciun 3984
This theorem is referenced by:  iununi  4065  abianfplem  6554  oe1m  6627  oarec  6644  oelim2  6677  bnj1143  28567
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-v 2866  df-dif 3231  df-nul 3532  df-iun 3986
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