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Theorem riotav 6554
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 6549 . 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 3745 . . . 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 2959 . . . . . 6  |-  x  e. 
_V
43biantrur 493 . . . . 5  |-  ( ph  <->  ( x  e.  _V  /\  ph ) )
54iotabii 5440 . . . 4  |-  ( iota
x ph )  =  ( iota x ( x  e.  _V  /\  ph ) )
62, 5syl6reqr 2487 . . 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 2971 . . . . . 6  |-  ( E! x  e.  _V  ph  <->  E! x ph )
8 iotanul 5433 . . . . . 6  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
97, 8sylnbi 298 . . . . 5  |-  ( -.  E! x  e.  _V  ph 
->  ( iota x ph )  =  (/) )
10 abid2 2553 . . . . . . 7  |-  { x  |  x  e.  _V }  =  _V
1110fveq2i 5731 . . . . . 6  |-  ( Undef `  { x  |  x  e.  _V } )  =  ( Undef `  _V )
12 vprc 4341 . . . . . . 7  |-  -.  _V  e.  _V
13 fvprc 5722 . . . . . . 7  |-  ( -. 
_V  e.  _V  ->  (
Undef `  _V )  =  (/) )
1412, 13ax-mp 8 . . . . . 6  |-  ( Undef `  _V )  =  (/)
1511, 14eqtri 2456 . . . . 5  |-  ( Undef `  { x  |  x  e.  _V } )  =  (/)
169, 15syl6eqr 2486 . . . 4  |-  ( -.  E! x  e.  _V  ph 
->  ( iota x ph )  =  ( Undef `  { x  |  x  e.  _V } ) )
17 iffalse 3746 . . . 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 2471 . . 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 158 . 2  |-  ( iota
x ph )  =  if ( E! x  e. 
_V  ph ,  ( iota
x ( x  e. 
_V  /\  ph ) ) ,  ( Undef `  {
x  |  x  e. 
_V } ) )
201, 19eqtr4i 2459 1  |-  ( iota_ x  e.  _V ph )  =  ( iota x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1652    e. wcel 1725   E!weu 2281   {cab 2422   E!wreu 2707   _Vcvv 2956   (/)c0 3628   ifcif 3739   iotacio 5416   ` cfv 5454   Undefcund 6541   iota_crio 6542
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-riota 6549
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