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Theorem riotauni 6238
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 2811 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
2 iotauni 5546 . . 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 6232 . 2  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
5 df-rab 2813 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
65unieqi 4244 . 2  |-  U. {
x  e.  A  |  ph }  =  U. {
x  |  ( x  e.  A  /\  ph ) }
73, 4, 63eqtr4g 2520 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 367    = wceq 1398    e. wcel 1823   E!weu 2284   {cab 2439   E!wreu 2806   {crab 2808   U.cuni 4235   iotacio 5532   iota_crio 6231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-un 3466  df-sn 4017  df-pr 4019  df-uni 4236  df-iota 5534  df-riota 6232
This theorem is referenced by:  riotassuniOLD  6268  supval2  7906  dfac2a  8501
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