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Theorem isomni 6976
Description: The predicate of being omniscient. (Contributed by Jim Kingdon, 28-Jun-2022.)
Assertion
Ref Expression
isomni  |-  ( A  e.  V  ->  ( A  e. Omni  <->  A. f ( f : A --> 2o  ->  ( E. x  e.  A  ( f `  x
)  =  (/)  \/  A. x  e.  A  (
f `  x )  =  1o ) ) ) )
Distinct variable group:    A, f, x
Allowed substitution hints:    V( x, f)

Proof of Theorem isomni
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 feq2 5226 . . . 4  |-  ( y  =  A  ->  (
f : y --> 2o  <->  f : A --> 2o ) )
2 rexeq 2604 . . . . 5  |-  ( y  =  A  ->  ( E. x  e.  y 
( f `  x
)  =  (/)  <->  E. x  e.  A  ( f `  x )  =  (/) ) )
3 raleq 2603 . . . . 5  |-  ( y  =  A  ->  ( A. x  e.  y 
( f `  x
)  =  1o  <->  A. x  e.  A  ( f `  x )  =  1o ) )
42, 3orbi12d 767 . . . 4  |-  ( y  =  A  ->  (
( E. x  e.  y  ( f `  x )  =  (/)  \/ 
A. x  e.  y  ( f `  x
)  =  1o )  <-> 
( E. x  e.  A  ( f `  x )  =  (/)  \/ 
A. x  e.  A  ( f `  x
)  =  1o ) ) )
51, 4imbi12d 233 . . 3  |-  ( y  =  A  ->  (
( f : y --> 2o  ->  ( E. x  e.  y  (
f `  x )  =  (/)  \/  A. x  e.  y  ( f `  x )  =  1o ) )  <->  ( f : A --> 2o  ->  ( E. x  e.  A  ( f `  x
)  =  (/)  \/  A. x  e.  A  (
f `  x )  =  1o ) ) ) )
65albidv 1780 . 2  |-  ( y  =  A  ->  ( A. f ( f : y --> 2o  ->  ( E. x  e.  y 
( f `  x
)  =  (/)  \/  A. x  e.  y  (
f `  x )  =  1o ) )  <->  A. f
( f : A --> 2o  ->  ( E. x  e.  A  ( f `  x )  =  (/)  \/ 
A. x  e.  A  ( f `  x
)  =  1o ) ) ) )
7 df-omni 6974 . 2  |- Omni  =  {
y  |  A. f
( f : y --> 2o  ->  ( E. x  e.  y  (
f `  x )  =  (/)  \/  A. x  e.  y  ( f `  x )  =  1o ) ) }
86, 7elab2g 2804 1  |-  ( A  e.  V  ->  ( A  e. Omni  <->  A. f ( f : A --> 2o  ->  ( E. x  e.  A  ( f `  x
)  =  (/)  \/  A. x  e.  A  (
f `  x )  =  1o ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 682   A.wal 1314    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   (/)c0 3333   -->wf 5089   ` cfv 5093   1oc1o 6274   2oc2o 6275  Omnicomni 6972
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-fn 5096  df-f 5097  df-omni 6974
This theorem is referenced by:  isomnimap  6977  finomni  6980  exmidomniim  6981  exmidomni  6982  omnimkv  6998
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