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Theorem riotaund 5603
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 5569 . 2  |-  ( iota_ x  e.  A  ph )  =  ( iota x
( x  e.  A  /\  ph ) )
2 df-reu 2362 . . 3  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
3 iotanul 4961 . . 3  |-  ( -.  E! x ( x  e.  A  /\  ph )  ->  ( iota x
( x  e.  A  /\  ph ) )  =  (/) )
42, 3sylnbi 636 . 2  |-  ( -.  E! x  e.  A  ph 
->  ( iota x ( x  e.  A  /\  ph ) )  =  (/) )
51, 4syl5eq 2129 1  |-  ( -.  E! x  e.  A  ph 
->  ( iota_ x  e.  A  ph )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1287    e. wcel 1436   E!weu 1945   E!wreu 2357   (/)c0 3275   iotacio 4944   iota_crio 5568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-reu 2362  df-v 2617  df-dif 2990  df-in 2994  df-ss 3001  df-nul 3276  df-sn 3437  df-uni 3637  df-iota 4946  df-riota 5569
This theorem is referenced by: (None)
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