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Theorem riotaundb 6362
Description: Restricted iota equals the undefined value of its domain of discourse  A when not meaningful. (Contributed by NM, 26-Sep-2011.)
Hypothesis
Ref Expression
riotaclb.1  |-  A  e. 
_V
Assertion
Ref Expression
riotaundb  |-  ( -.  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 riotaundb
StepHypRef Expression
1 riotaund 6357 . 2  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A ph )  =  ( Undef `  A ) )
2 riotacl 6335 . . . 4  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  A
)
3 riotaclb.1 . . . . 5  |-  A  e. 
_V
4 undefnel2 6318 . . . . 5  |-  ( A  e.  _V  ->  -.  ( Undef `  A )  e.  A )
53, 4ax-mp 8 . . . 4  |-  -.  ( Undef `  A )  e.  A
6 nelne2 2549 . . . 4  |-  ( ( ( iota_ x  e.  A ph )  e.  A  /\  -.  ( Undef `  A
)  e.  A )  ->  ( iota_ x  e.  A ph )  =/=  ( Undef `  A )
)
72, 5, 6sylancl 643 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =/=  ( Undef `  A ) )
87necon2bi 2505 . 2  |-  ( (
iota_ x  e.  A ph )  =  ( Undef `  A )  ->  -.  E! x  e.  A  ph )
91, 8impbii 180 1  |-  ( -.  E! x  e.  A  ph  <->  (
iota_ x  e.  A ph )  =  ( Undef `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1632    e. wcel 1696    =/= wne 2459   E!wreu 2558   _Vcvv 2801   ` cfv 5271   Undefcund 6312   iota_crio 6313
This theorem is referenced by:  frgra2v  28423
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  ax-pr 4230  ax-un 4528
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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-undef 6314  df-riota 6320
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