MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  riotav Unicode version

Theorem riotav 6325
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 6320 . 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 3584 . . . 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 2804 . . . . . 6  |-  x  e. 
_V
43biantrur 492 . . . . 5  |-  ( ph  <->  ( x  e.  _V  /\  ph ) )
54iotabii 5257 . . . 4  |-  ( iota
x ph )  =  ( iota x ( x  e.  _V  /\  ph ) )
62, 5syl6reqr 2347 . . 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 2816 . . . . . 6  |-  ( E! x  e.  _V  ph  <->  E! x ph )
8 iotanul 5250 . . . . . 6  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
97, 8sylnbi 297 . . . . 5  |-  ( -.  E! x  e.  _V  ph 
->  ( iota x ph )  =  (/) )
10 abid2 2413 . . . . . . 7  |-  { x  |  x  e.  _V }  =  _V
1110fveq2i 5544 . . . . . 6  |-  ( Undef `  { x  |  x  e.  _V } )  =  ( Undef `  _V )
12 vprc 4168 . . . . . . 7  |-  -.  _V  e.  _V
13 fvprc 5535 . . . . . . 7  |-  ( -. 
_V  e.  _V  ->  (
Undef `  _V )  =  (/) )
1412, 13ax-mp 8 . . . . . 6  |-  ( Undef `  _V )  =  (/)
1511, 14eqtri 2316 . . . . 5  |-  ( Undef `  { x  |  x  e.  _V } )  =  (/)
169, 15syl6eqr 2346 . . . 4  |-  ( -.  E! x  e.  _V  ph 
->  ( iota x ph )  =  ( Undef `  { x  |  x  e.  _V } ) )
17 iffalse 3585 . . . 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 2331 . . 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 2319 1  |-  ( iota_ x  e.  _V ph )  =  ( iota x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1632    e. wcel 1696   E!weu 2156   {cab 2282   E!wreu 2558   _Vcvv 2801   (/)c0 3468   ifcif 3578   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-sep 4157  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
  Copyright terms: Public domain W3C validator