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Theorem riotaund 6196
Description: Restricted iota equals the empty set when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 13-Sep-2018.)
Assertion
Ref Expression
riotaund  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A  ph )  =  (/) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem riotaund
StepHypRef Expression
1 df-riota 6160 . 2  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
2 df-reu 2805 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 iotanul 5503 . . 3  |-  ( -.  E! x ( x  e.  A  /\  ph )  ->  ( iota x
( x  e.  A  /\  ph ) )  =  (/) )
42, 3sylnbi 306 . 2  |-  ( -.  E! x  e.  A  ph 
->  ( iota x ( x  e.  A  /\  ph ) )  =  (/) )
51, 4syl5eq 2507 1  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A  ph )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E!weu 2262   E!wreu 2800   (/)c0 3744   iotacio 5486   iota_crio 6159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-v 3078  df-dif 3438  df-in 3442  df-ss 3449  df-nul 3745  df-sn 3985  df-uni 4199  df-iota 5488  df-riota 6160
This theorem is referenced by:  riotassuniOLD  6197  riotaclb  6198  supval2  7815  lubval  15272  glbval  15285
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