Theorem isomni 7299

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.

Step	Hyp	Ref	Expression
1		feq2 5456	. . . 4 |- ( y = A -> ( f : y --> 2o <-> f : A --> 2o ) )
2		rexeq 2729	. . . . 5 |- ( y = A -> ( E. x e. y ( f ` x ) = (/) <-> E. x e. A ( f ` x ) = (/) ) )
3		raleq 2728	. . . . 5 |- ( y = A -> ( A. x e. y ( f ` x ) = 1o <-> A. x e. A ( f ` x ) = 1o ) )
4	2, 3	orbi12d 798	. . . 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 ) ) )
5	1, 4	imbi12d 234	. . 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 ) ) ) )
6	5	albidv 1870	. 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 7298	. 2 |- Omni = { y | A. f ( f : y --> 2o -> ( E. x e. y ( f ` x ) = (/) \/ A. x e. y ( f ` x ) = 1o ) ) }
8	6, 7	elab2g 2950	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 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 713   A.wal 1393   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   (/)c0 3491   -->wf 5313   ` cfv 5317   1oc1o 6553   2oc2o 6554  Omnicomni 7297

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-fn 5320  df-f 5321  df-omni 7298

This theorem is referenced by:  isomnimap  7300  finomni  7303  exmidomniim  7304  exmidomni  7305  omnimkv  7319  omniwomnimkv  7330  nninfctlemfo  12556