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Theorem iswomni 7162
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 5349 . . . 4  |-  ( y  =  A  ->  (
f : y --> 2o  <->  f : A --> 2o ) )
2 raleq 2672 . . . . 5  |-  ( y  =  A  ->  ( A. x  e.  y 
( f `  x
)  =  1o  <->  A. x  e.  A  ( f `  x )  =  1o ) )
32dcbid 838 . . . 4  |-  ( y  =  A  ->  (DECID  A. x  e.  y  (
f `  x )  =  1o  <-> DECID  A. x  e.  A  ( f `  x )  =  1o ) )
41, 3imbi12d 234 . . 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 1824 . 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 7161 . 2  |- WOmni  =  {
y  |  A. f
( f : y --> 2o  -> DECID  A. x  e.  y  ( f `  x
)  =  1o ) }
75, 6elab2g 2884 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 105  DECID wdc 834   A.wal 1351    = wceq 1353    e. wcel 2148   A.wral 2455   -->wf 5212   ` cfv 5216   1oc1o 6409   2oc2o 6410  WOmnicwomni 7160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-fn 5219  df-f 5220  df-womni 7161
This theorem is referenced by:  iswomnimap  7163  omniwomnimkv  7164
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