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Theorem isomni 7001
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 5251 . . . 4  |-  ( y  =  A  ->  (
f : y --> 2o  <->  f : A --> 2o ) )
2 rexeq 2625 . . . . 5  |-  ( y  =  A  ->  ( E. x  e.  y 
( f `  x
)  =  (/)  <->  E. x  e.  A  ( f `  x )  =  (/) ) )
3 raleq 2624 . . . . 5  |-  ( y  =  A  ->  ( A. x  e.  y 
( f `  x
)  =  1o  <->  A. x  e.  A  ( f `  x )  =  1o ) )
42, 3orbi12d 782 . . . 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 1796 . 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 6999 . 2  |- Omni  =  {
y  |  A. f
( f : y --> 2o  ->  ( E. x  e.  y  (
f `  x )  =  (/)  \/  A. x  e.  y  ( f `  x )  =  1o ) ) }
86, 7elab2g 2826 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 697   A.wal 1329    = wceq 1331    e. wcel 1480   A.wral 2414   E.wrex 2415   (/)c0 3358   -->wf 5114   ` cfv 5118   1oc1o 6299   2oc2o 6300  Omnicomni 6997
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-fn 5121  df-f 5122  df-omni 6999
This theorem is referenced by:  isomnimap  7002  finomni  7005  exmidomniim  7006  exmidomni  7007  omnimkv  7023
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