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Theorem riotaund 5772
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 5738 . 2  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
2 df-reu 2424 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 iotanul 5111 . . 3  |-  ( -.  E! x ( x  e.  A  /\  ph )  ->  ( iota x
( x  e.  A  /\  ph ) )  =  (/) )
42, 3sylnbi 668 . 2  |-  ( -.  E! x  e.  A  ph 
->  ( iota x ( x  e.  A  /\  ph ) )  =  (/) )
51, 4syl5eq 2185 1  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A  ph )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   E!weu 2000   E!wreu 2419   (/)c0 3368   iotacio 5094   iota_crio 5737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-nul 3369  df-sn 3538  df-uni 3745  df-iota 5096  df-riota 5738
This theorem is referenced by: (None)
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