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Theorem riotauni 6523
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 6522 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
2 df-reu 2681 . . . 4  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 iotauni 5397 . . . 4  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( iota x ( x  e.  A  /\  ph ) )  =  U. { x  |  (
x  e.  A  /\  ph ) } )
42, 3sylbi 188 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota x ( x  e.  A  /\  ph ) )  =  U. { x  |  (
x  e.  A  /\  ph ) } )
5 df-rab 2683 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
65unieqi 3993 . . 3  |-  U. {
x  e.  A  |  ph }  =  U. {
x  |  ( x  e.  A  /\  ph ) }
74, 6syl6eqr 2462 . 2  |-  ( E! x  e.  A  ph  ->  ( iota x ( x  e.  A  /\  ph ) )  =  U. { x  e.  A  |  ph } )
81, 7eqtrd 2444 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 359    = wceq 1649    e. wcel 1721   E!weu 2262   {cab 2398   E!wreu 2676   {crab 2678   U.cuni 3983   iotacio 5383   iota_crio 6509
This theorem is referenced by:  riotassuni  6555  supval2  7424  dfac2a  7974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-un 3293  df-if 3708  df-sn 3788  df-pr 3789  df-uni 3984  df-iota 5385  df-riota 6516
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