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Theorem iunconst 4103
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 3724 . . 3  |-  ( A  =/=  (/)  ->  ( y  e.  B  <->  E. x  e.  A  y  e.  B )
)
2 eliun 4099 . . 3  |-  ( y  e.  U_ x  e.  A  B  <->  E. x  e.  A  y  e.  B )
31, 2syl6rbbr 257 . 2  |-  ( A  =/=  (/)  ->  ( y  e.  U_ x  e.  A  B 
<->  y  e.  B ) )
43eqrdv 2436 1  |-  ( A  =/=  (/)  ->  U_ x  e.  A  B  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   (/)c0 3630   U_ciun 4095
This theorem is referenced by:  iununi  4178  abianfplem  6718  oe1m  6791  oarec  6808  oelim2  6841  mblfinlem2  26256  bnj1143  29235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-dif 3325  df-nul 3631  df-iun 4097
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