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

Theorem riotauni 6162
Description: Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)
Assertion
Ref Expression
riotauni  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  =  U. {
x  e.  A  |  ph } )

Proof of Theorem riotauni
StepHypRef Expression
1 df-reu 2803 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
2 iotauni 5496 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( iota x ( x  e.  A  /\  ph ) )  =  U. { x  |  (
x  e.  A  /\  ph ) } )
31, 2sylbi 195 . 2  |-  ( E! x  e.  A  ph  ->  ( iota x ( x  e.  A  /\  ph ) )  =  U. { x  |  (
x  e.  A  /\  ph ) } )
4 df-riota 6156 . 2  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
5 df-rab 2805 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
65unieqi 4203 . 2  |-  U. {
x  e.  A  |  ph }  =  U. {
x  |  ( x  e.  A  /\  ph ) }
73, 4, 63eqtr4g 2518 1  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  =  U. {
x  e.  A  |  ph } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E!weu 2261   {cab 2437   E!wreu 2798   {crab 2800   U.cuni 4194   iotacio 5482   iota_crio 6155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-un 3436  df-sn 3981  df-pr 3983  df-uni 4195  df-iota 5484  df-riota 6156
This theorem is referenced by:  riotassuniOLD  6193  supval2  7811  dfac2a  8405
  Copyright terms: Public domain W3C validator