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Theorem dfiun2 3953
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 3951 . 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 2625 1  |-  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B }
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801   U.cuni 3843   U_ciun 3921
This theorem is referenced by:  funcnvuni  5333  fun11iun  5509  fniunfv  5789  tfrlem8  6416  rdglim2a  6462  rankuni  7551  cardiun  7631  kmlem11  7802  cfslb2n  7910  enfin2i  7963  pwcfsdom  8221  rankcf  8415  tskuni  8421  discmp  17141  cmpsublem  17142  cmpsub  17143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-uni 3844  df-iun 3923
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