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Theorem dfiun2 4199
Description: Alternate definition of indexed union when  B is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 27-Jun-1998.) (Revised by David Abernethy, 19-Jun-2012.)
Hypothesis
Ref Expression
dfiun2.1  |-  B  e. 
_V
Assertion
Ref Expression
dfiun2  |-  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B }
Distinct variable groups:    x, y    y, A    y, B
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem dfiun2
StepHypRef Expression
1 dfiun2g 4197 . 2  |-  ( A. x  e.  A  B  e.  _V  ->  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B } )
2 dfiun2.1 . . 3  |-  B  e. 
_V
32a1i 11 . 2  |-  ( x  e.  A  ->  B  e.  _V )
41, 3mprg 2780 1  |-  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756   {cab 2424   E.wrex 2711   _Vcvv 2967   U.cuni 4086   U_ciun 4166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2715  df-rex 2716  df-v 2969  df-uni 4087  df-iun 4168
This theorem is referenced by:  fniunfv  5959  funcnvuni  6525  fun11iun  6532  tfrlem8  6835  rdglim2a  6881  rankuni  8062  cardiun  8144  kmlem11  8321  cfslb2n  8429  enfin2i  8482  pwcfsdom  8739  rankcf  8936  tskuni  8942  discmp  18976  cmpsublem  18977  cmpsub  18978
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