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Theorem riotaprc 6579
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 5714 . . . . 5  |-  ( -.  A  e.  _V  ->  (
Undef `  A )  =  (/) )
21adantr 452 . . . 4  |-  ( ( -.  A  e.  _V  /\ 
-.  E! x  e.  A  ph )  -> 
( Undef `  A )  =  (/) )
3 riotaund 6578 . . . . 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 2704 . . . . . 6  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
6 iotanul 5425 . . . . . 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 2477 . . 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 6547 . 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 1652    e. wcel 1725   E!weu 2280   E!wreu 2699   _Vcvv 2948   (/)c0 3620   iotacio 5408   ` cfv 5446   Undefcund 6533   iota_crio 6534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330  ax-pow 4369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-riota 6541
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