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Theorem iswomni 7108
Description: The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.)
Assertion
Ref Expression
iswomni  |-  ( A  e.  V  ->  ( A  e. WOmni  <->  A. f ( f : A --> 2o  -> DECID  A. x  e.  A  ( f `  x )  =  1o ) ) )
Distinct variable group:    A, f, x
Allowed substitution hints:    V( x, f)

Proof of Theorem iswomni
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 feq2 5303 . . . 4  |-  ( y  =  A  ->  (
f : y --> 2o  <->  f : A --> 2o ) )
2 raleq 2652 . . . . 5  |-  ( y  =  A  ->  ( A. x  e.  y 
( f `  x
)  =  1o  <->  A. x  e.  A  ( f `  x )  =  1o ) )
32dcbid 824 . . . 4  |-  ( y  =  A  ->  (DECID  A. x  e.  y  (
f `  x )  =  1o  <-> DECID  A. x  e.  A  ( f `  x )  =  1o ) )
41, 3imbi12d 233 . . 3  |-  ( y  =  A  ->  (
( f : y --> 2o  -> DECID  A. x  e.  y  ( f `  x
)  =  1o )  <-> 
( f : A --> 2o  -> DECID  A. x  e.  A  ( f `  x
)  =  1o ) ) )
54albidv 1804 . 2  |-  ( y  =  A  ->  ( A. f ( f : y --> 2o  -> DECID  A. x  e.  y  ( f `  x
)  =  1o )  <->  A. f ( f : A --> 2o  -> DECID  A. x  e.  A  ( f `  x
)  =  1o ) ) )
6 df-womni 7107 . 2  |- WOmni  =  {
y  |  A. f
( f : y --> 2o  -> DECID  A. x  e.  y  ( f `  x
)  =  1o ) }
75, 6elab2g 2859 1  |-  ( A  e.  V  ->  ( A  e. WOmni  <->  A. f ( f : A --> 2o  -> DECID  A. x  e.  A  ( f `  x )  =  1o ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104  DECID wdc 820   A.wal 1333    = wceq 1335    e. wcel 2128   A.wral 2435   -->wf 5166   ` cfv 5170   1oc1o 6356   2oc2o 6357  WOmnicwomni 7106
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-fn 5173  df-f 5174  df-womni 7107
This theorem is referenced by:  iswomnimap  7109  omniwomnimkv  7110
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