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Theorem riotaprc 6358
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 5535 . . . . 5  |-  ( -.  A  e.  _V  ->  (
Undef `  A )  =  (/) )
21adantr 451 . . . 4  |-  ( ( -.  A  e.  _V  /\ 
-.  E! x  e.  A  ph )  -> 
( Undef `  A )  =  (/) )
3 riotaund 6357 . . . . 5  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A ph )  =  ( Undef `  A ) )
43adantl 452 . . . 4  |-  ( ( -.  A  e.  _V  /\ 
-.  E! x  e.  A  ph )  -> 
( iota_ x  e.  A ph )  =  ( Undef `  A ) )
5 df-reu 2563 . . . . . 6  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
6 iotanul 5250 . . . . . 6  |-  ( -.  E! x ( x  e.  A  /\  ph )  ->  ( iota x
( x  e.  A  /\  ph ) )  =  (/) )
75, 6sylnbi 297 . . . . 5  |-  ( -.  E! x  e.  A  ph 
->  ( iota x ( x  e.  A  /\  ph ) )  =  (/) )
87adantl 452 . . . 4  |-  ( ( -.  A  e.  _V  /\ 
-.  E! x  e.  A  ph )  -> 
( iota x ( x  e.  A  /\  ph ) )  =  (/) )
92, 4, 83eqtr4d 2338 . . 3  |-  ( ( -.  A  e.  _V  /\ 
-.  E! x  e.  A  ph )  -> 
( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
109ex 423 . 2  |-  ( -.  A  e.  _V  ->  ( -.  E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) ) )
11 riotaiota 6326 . 2  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =  ( iota x ( x  e.  A  /\  ph )
) )
1210, 11pm2.61d2 152 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 358    = wceq 1632    e. wcel 1696   E!weu 2156   E!wreu 2558   _Vcvv 2801   (/)c0 3468   iotacio 5233   ` cfv 5271   Undefcund 6312   iota_crio 6313
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-riota 6320
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