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Theorem riotaund 6524
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 3691 . 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 6487 . 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 2506 . . . 4  |-  { x  |  x  e.  A }  =  A
43eqcomi 2393 . . 3  |-  A  =  { x  |  x  e.  A }
54fveq2i 5673 . 2  |-  ( Undef `  A )  =  (
Undef `  { x  |  x  e.  A }
)
61, 2, 53eqtr4g 2446 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 359    = wceq 1649    e. wcel 1717   {cab 2375   E!wreu 2653   ifcif 3684   iotacio 5358   ` cfv 5396   Undefcund 6479   iota_crio 6480
This theorem is referenced by:  riotaprc  6525  riotassuni  6526  riotaclbg  6527  riotaundb  6529
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-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
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-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-iota 5360  df-fv 5404  df-riota 6487
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