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

Theorem riotauni 6311
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 riotaiota 6310 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
2 df-reu 2550 . . . 4  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 iotauni 5231 . . . 4  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( iota x ( x  e.  A  /\  ph ) )  =  U. { x  |  (
x  e.  A  /\  ph ) } )
42, 3sylbi 187 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota x ( x  e.  A  /\  ph ) )  =  U. { x  |  (
x  e.  A  /\  ph ) } )
5 df-rab 2552 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
65unieqi 3837 . . 3  |-  U. {
x  e.  A  |  ph }  =  U. {
x  |  ( x  e.  A  /\  ph ) }
74, 6syl6eqr 2333 . 2  |-  ( E! x  e.  A  ph  ->  ( iota x ( x  e.  A  /\  ph ) )  =  U. { x  e.  A  |  ph } )
81, 7eqtrd 2315 1  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  U. { x  e.  A  |  ph } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E!weu 2143   {cab 2269   E!wreu 2545   {crab 2547   U.cuni 3827   iotacio 5217   iota_crio 6297
This theorem is referenced by:  riotassuni  6343  supval2  7206  dfac2a  7756  supexr  25631
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-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-un 3157  df-if 3566  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219  df-riota 6304
  Copyright terms: Public domain W3C validator