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Theorem riotav 6309
Description: An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.)
Assertion
Ref Expression
riotav  |-  ( iota_ x  e.  _V ph )  =  ( iota x ph )

Proof of Theorem riotav
StepHypRef Expression
1 df-riota 6304 . 2  |-  ( iota_ x  e.  _V ph )  =  if ( E! x  e.  _V  ph ,  ( iota x ( x  e.  _V  /\  ph ) ) ,  (
Undef `  { x  |  x  e.  _V }
) )
2 iftrue 3571 . . . 4  |-  ( E! x  e.  _V  ph  ->  if ( E! x  e.  _V  ph ,  ( iota x ( x  e.  _V  /\  ph ) ) ,  (
Undef `  { x  |  x  e.  _V }
) )  =  ( iota x ( x  e.  _V  /\  ph ) ) )
3 vex 2791 . . . . . 6  |-  x  e. 
_V
43biantrur 492 . . . . 5  |-  ( ph  <->  ( x  e.  _V  /\  ph ) )
54iotabii 5241 . . . 4  |-  ( iota
x ph )  =  ( iota x ( x  e.  _V  /\  ph ) )
62, 5syl6reqr 2334 . . 3  |-  ( E! x  e.  _V  ph  ->  ( iota x ph )  =  if ( E! x  e.  _V  ph ,  ( iota x
( x  e.  _V  /\ 
ph ) ) ,  ( Undef `  { x  |  x  e.  _V } ) ) )
7 reuv 2803 . . . . . 6  |-  ( E! x  e.  _V  ph  <->  E! x ph )
8 iotanul 5234 . . . . . 6  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
97, 8sylnbi 297 . . . . 5  |-  ( -.  E! x  e.  _V  ph 
->  ( iota x ph )  =  (/) )
10 abid2 2400 . . . . . . 7  |-  { x  |  x  e.  _V }  =  _V
1110fveq2i 5528 . . . . . 6  |-  ( Undef `  { x  |  x  e.  _V } )  =  ( Undef `  _V )
12 vprc 4152 . . . . . . 7  |-  -.  _V  e.  _V
13 fvprc 5519 . . . . . . 7  |-  ( -. 
_V  e.  _V  ->  (
Undef `  _V )  =  (/) )
1412, 13ax-mp 8 . . . . . 6  |-  ( Undef `  _V )  =  (/)
1511, 14eqtri 2303 . . . . 5  |-  ( Undef `  { x  |  x  e.  _V } )  =  (/)
169, 15syl6eqr 2333 . . . 4  |-  ( -.  E! x  e.  _V  ph 
->  ( iota x ph )  =  ( Undef `  { x  |  x  e.  _V } ) )
17 iffalse 3572 . . . 4  |-  ( -.  E! x  e.  _V  ph 
->  if ( E! x  e.  _V  ph ,  ( iota x ( x  e.  _V  /\  ph ) ) ,  (
Undef `  { x  |  x  e.  _V }
) )  =  (
Undef `  { x  |  x  e.  _V }
) )
1816, 17eqtr4d 2318 . . 3  |-  ( -.  E! x  e.  _V  ph 
->  ( iota x ph )  =  if ( E! x  e.  _V  ph ,  ( iota x
( x  e.  _V  /\ 
ph ) ) ,  ( Undef `  { x  |  x  e.  _V } ) ) )
196, 18pm2.61i 156 . 2  |-  ( iota
x ph )  =  if ( E! x  e. 
_V  ph ,  ( iota
x ( x  e. 
_V  /\  ph ) ) ,  ( Undef `  {
x  |  x  e. 
_V } ) )
201, 19eqtr4i 2306 1  |-  ( iota_ x  e.  _V ph )  =  ( iota x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684   E!weu 2143   {cab 2269   E!wreu 2545   _Vcvv 2788   (/)c0 3455   ifcif 3565   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-sep 4141  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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