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Theorem riotaundb 6583
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 6578 . 2  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A ph )  =  ( Undef `  A ) )
2 riotacl 6556 . . . 4  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  A
)
3 riotaclb.1 . . . . 5  |-  A  e. 
_V
4 undefnel2 6539 . . . . 5  |-  ( A  e.  _V  ->  -.  ( Undef `  A )  e.  A )
53, 4ax-mp 8 . . . 4  |-  -.  ( Undef `  A )  e.  A
6 nelne2 2688 . . . 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 2644 . 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 1652    e. wcel 1725    =/= wne 2598   E!wreu 2699   _Vcvv 2948   ` cfv 5446   Undefcund 6533   iota_crio 6534
This theorem is referenced by:  frgra2v  28326
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-undef 6535  df-riota 6541
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