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Theorem isomni 6790
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 5146 . . . 4  |-  ( y  =  A  ->  (
f : y --> 2o  <->  f : A --> 2o ) )
2 rexeq 2563 . . . . 5  |-  ( y  =  A  ->  ( E. x  e.  y 
( f `  x
)  =  (/)  <->  E. x  e.  A  ( f `  x )  =  (/) ) )
3 raleq 2562 . . . . 5  |-  ( y  =  A  ->  ( A. x  e.  y 
( f `  x
)  =  1o  <->  A. x  e.  A  ( f `  x )  =  1o ) )
42, 3orbi12d 742 . . . 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 232 . . 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 1752 . 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 6788 . 2  |- Omni  =  {
y  |  A. f
( f : y --> 2o  ->  ( E. x  e.  y  (
f `  x )  =  (/)  \/  A. x  e.  y  ( f `  x )  =  1o ) ) }
86, 7elab2g 2762 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 103    \/ wo 664   A.wal 1287    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   (/)c0 3286   -->wf 5011   ` cfv 5015   1oc1o 6174   2oc2o 6175  Omnicomni 6786
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-fn 5018  df-f 5019  df-omni 6788
This theorem is referenced by:  isomnimap  6791  finomni  6794  exmidomniim  6795  exmidomni  6796
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