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Theorem riotauni 6269
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 2782 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
2 iotauni 5573 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( iota x ( x  e.  A  /\  ph ) )  =  U. { x  |  (
x  e.  A  /\  ph ) } )
31, 2sylbi 198 . 2  |-  ( E! x  e.  A  ph  ->  ( iota x ( x  e.  A  /\  ph ) )  =  U. { x  |  (
x  e.  A  /\  ph ) } )
4 df-riota 6263 . 2  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
5 df-rab 2784 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
65unieqi 4225 . 2  |-  U. {
x  e.  A  |  ph }  =  U. {
x  |  ( x  e.  A  /\  ph ) }
73, 4, 63eqtr4g 2488 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 370    = wceq 1437    e. wcel 1868   E!weu 2265   {cab 2407   E!wreu 2777   {crab 2779   U.cuni 4216   iotacio 5559   iota_crio 6262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-un 3441  df-sn 3997  df-pr 3999  df-uni 4217  df-iota 5561  df-riota 6263
This theorem is referenced by:  riotassuni  6299  supval2  7971  dfac2a  8560
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