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Theorem dfiun2 4303
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 4301 . 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 2770 1  |-  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452    e. wcel 1904   {cab 2457   E.wrex 2757   _Vcvv 3031   U.cuni 4190   U_ciun 4269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-v 3033  df-uni 4191  df-iun 4271
This theorem is referenced by:  fniunfv  6170  funcnvuni  6765  fun11iun  6772  tfrlem8  7120  rdglim2a  7169  rankuni  8352  cardiun  8434  kmlem11  8608  cfslb2n  8716  enfin2i  8769  pwcfsdom  9026  rankcf  9220  tskuni  9226  discmp  20490  cmpsublem  20491  cmpsub  20492
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