MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfiun2 Structured version   Unicode version

Theorem dfiun2 4352
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 4350 . 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 2820 1  |-  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    e. wcel 1762   {cab 2445   E.wrex 2808   _Vcvv 3106   U.cuni 4238   U_ciun 4318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-v 3108  df-uni 4239  df-iun 4320
This theorem is referenced by:  fniunfv  6138  funcnvuni  6727  fun11iun  6734  tfrlem8  7043  rdglim2a  7089  rankuni  8270  cardiun  8352  kmlem11  8529  cfslb2n  8637  enfin2i  8690  pwcfsdom  8947  rankcf  9144  tskuni  9150  discmp  19657  cmpsublem  19658  cmpsub  19659
  Copyright terms: Public domain W3C validator