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Theorem riotaund 6357
Description: Restricted iota equals the undefined value of its domain of discourse  A when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)
Assertion
Ref Expression
riotaund  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A ph )  =  ( Undef `  A ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem riotaund
StepHypRef Expression
1 iffalse 3585 . 2  |-  ( -.  E! x  e.  A  ph 
->  if ( E! x  e.  A  ph ,  ( iota x ( x  e.  A  /\  ph ) ) ,  (
Undef `  { x  |  x  e.  A }
) )  =  (
Undef `  { x  |  x  e.  A }
) )
2 df-riota 6320 . 2  |-  ( iota_ x  e.  A ph )  =  if ( E! x  e.  A  ph ,  ( iota x ( x  e.  A  /\  ph ) ) ,  (
Undef `  { x  |  x  e.  A }
) )
3 abid2 2413 . . . 4  |-  { x  |  x  e.  A }  =  A
43eqcomi 2300 . . 3  |-  A  =  { x  |  x  e.  A }
54fveq2i 5544 . 2  |-  ( Undef `  A )  =  (
Undef `  { x  |  x  e.  A }
)
61, 2, 53eqtr4g 2353 1  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A ph )  =  ( Undef `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   E!wreu 2558   ifcif 3578   iotacio 5233   ` cfv 5271   Undefcund 6312   iota_crio 6313
This theorem is referenced by:  riotaprc  6358  riotassuni  6359  riotaclbg  6360  riotaundb  6362
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  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
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