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Theorem riotauni 5499
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 2356 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
2 iotauni 4903 . . 3  |-  ( E! x ( x  e.  A  /\  ph )  ->  ( iota x ( x  e.  A  /\  ph ) )  =  U. { x  |  (
x  e.  A  /\  ph ) } )
31, 2sylbi 119 . 2  |-  ( E! x  e.  A  ph  ->  ( iota x ( x  e.  A  /\  ph ) )  =  U. { x  |  (
x  e.  A  /\  ph ) } )
4 df-riota 5493 . 2  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
5 df-rab 2358 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
65unieqi 3613 . 2  |-  U. {
x  e.  A  |  ph }  =  U. {
x  |  ( x  e.  A  /\  ph ) }
73, 4, 63eqtr4g 2139 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 102    = wceq 1285    e. wcel 1434   E!weu 1942   {cab 2068   E!wreu 2351   {crab 2353   U.cuni 3603   iotacio 4889   iota_crio 5492
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-un 2978  df-sn 3406  df-pr 3407  df-uni 3604  df-iota 4891  df-riota 5493
This theorem is referenced by:  supval2ti  6457
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