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Theorem riotaund 6193
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 6158 . 2  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
2 df-reu 2739 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 iotanul 5475 . . 3  |-  ( -.  E! x ( x  e.  A  /\  ph )  ->  ( iota x
( x  e.  A  /\  ph ) )  =  (/) )
42, 3sylnbi 304 . 2  |-  ( -.  E! x  e.  A  ph 
->  ( iota x ( x  e.  A  /\  ph ) )  =  (/) )
51, 4syl5eq 2435 1  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A  ph )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   E!weu 2218   E!wreu 2734   (/)c0 3711   iotacio 5458   iota_crio 6157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-v 3036  df-dif 3392  df-in 3396  df-ss 3403  df-nul 3712  df-sn 3945  df-uni 4164  df-iota 5460  df-riota 6158
This theorem is referenced by:  riotassuni  6194  riotaclb  6195  supval2  7829  lubval  15731  glbval  15744
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