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Theorem riotaprc 6342
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 5519 . . . . 5  |-  ( -.  A  e.  _V  ->  (
Undef `  A )  =  (/) )
21adantr 451 . . . 4  |-  ( ( -.  A  e.  _V  /\ 
-.  E! x  e.  A  ph )  -> 
( Undef `  A )  =  (/) )
3 riotaund 6341 . . . . 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 2550 . . . . . 6  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
6 iotanul 5234 . . . . . 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 2325 . . 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 6310 . 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 1623    e. wcel 1684   E!weu 2143   E!wreu 2545   _Vcvv 2788   (/)c0 3455   iotacio 5217   ` cfv 5255   Undefcund 6296   iota_crio 6297
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149  ax-pow 4188
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-riota 6304
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