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Theorem riotaundb 6527
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 6522 . 2  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A ph )  =  ( Undef `  A ) )
2 riotacl 6500 . . . 4  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  A
)
3 riotaclb.1 . . . . 5  |-  A  e. 
_V
4 undefnel2 6483 . . . . 5  |-  ( A  e.  _V  ->  -.  ( Undef `  A )  e.  A )
53, 4ax-mp 8 . . . 4  |-  -.  ( Undef `  A )  e.  A
6 nelne2 2640 . . . 4  |-  ( ( ( iota_ x  e.  A ph )  e.  A  /\  -.  ( Undef `  A
)  e.  A )  ->  ( iota_ x  e.  A ph )  =/=  ( Undef `  A )
)
72, 5, 6sylancl 644 . . 3  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  =/=  ( Undef `  A ) )
87necon2bi 2596 . 2  |-  ( (
iota_ x  e.  A ph )  =  ( Undef `  A )  ->  -.  E! x  e.  A  ph )
91, 8impbii 181 1  |-  ( -.  E! x  e.  A  ph  <->  (
iota_ x  e.  A ph )  =  ( Undef `  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    = wceq 1649    e. wcel 1717    =/= wne 2550   E!wreu 2651   _Vcvv 2899   ` cfv 5394   Undefcund 6477   iota_crio 6478
This theorem is referenced by:  frgra2v  27752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-undef 6479  df-riota 6485
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