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Theorem riotaprc 6523
Description: For proper classes, restricted and unrestricted iota are the same. (Contributed by NM, 15-Sep-2011.)
Assertion
Ref Expression
riotaprc  |-  ( -.  A  e.  _V  ->  (
iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem riotaprc
StepHypRef Expression
1 fvprc 5662 . . . . 5  |-  ( -.  A  e.  _V  ->  (
Undef `  A )  =  (/) )
21adantr 452 . . . 4  |-  ( ( -.  A  e.  _V  /\ 
-.  E! x  e.  A  ph )  -> 
( Undef `  A )  =  (/) )
3 riotaund 6522 . . . . 5  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A ph )  =  ( Undef `  A ) )
43adantl 453 . . . 4  |-  ( ( -.  A  e.  _V  /\ 
-.  E! x  e.  A  ph )  -> 
( iota_ x  e.  A ph )  =  ( Undef `  A ) )
5 df-reu 2656 . . . . . 6  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
6 iotanul 5373 . . . . . 6  |-  ( -.  E! x ( x  e.  A  /\  ph )  ->  ( iota x
( x  e.  A  /\  ph ) )  =  (/) )
75, 6sylnbi 298 . . . . 5  |-  ( -.  E! x  e.  A  ph 
->  ( iota x ( x  e.  A  /\  ph ) )  =  (/) )
87adantl 453 . . . 4  |-  ( ( -.  A  e.  _V  /\ 
-.  E! x  e.  A  ph )  -> 
( iota x ( x  e.  A  /\  ph ) )  =  (/) )
92, 4, 83eqtr4d 2429 . . 3  |-  ( ( -.  A  e.  _V  /\ 
-.  E! x  e.  A  ph )  -> 
( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
109ex 424 . 2  |-  ( -.  A  e.  _V  ->  ( -.  E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) ) )
11 riotaiota 6491 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
1210, 11pm2.61d2 154 1  |-  ( -.  A  e.  _V  ->  (
iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   E!weu 2238   E!wreu 2651   _Vcvv 2899   (/)c0 3571   iotacio 5356   ` cfv 5394   Undefcund 6477   iota_crio 6478
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-nul 4279  ax-pow 4318
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-riota 6485
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