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Theorem dfiun2 3937
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 3935 . 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 10 . 2  |-  ( x  e.  A  ->  B  e.  _V )
41, 3mprg 2612 1  |-  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B }
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   U.cuni 3827   U_ciun 3905
This theorem is referenced by:  funcnvuni  5317  fun11iun  5493  fniunfv  5773  tfrlem8  6400  rdglim2a  6446  rankuni  7535  cardiun  7615  kmlem11  7786  cfslb2n  7894  enfin2i  7947  pwcfsdom  8205  rankcf  8399  tskuni  8405  discmp  17125  cmpsublem  17126  cmpsub  17127
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-uni 3828  df-iun 3907
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