Theorem isomni 7100

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 5321	. . . 4 |- ( y = A -> ( f : y --> 2o <-> f : A --> 2o ) )
2		rexeq 2662	. . . . 5 |- ( y = A -> ( E. x e. y ( f ` x ) = (/) <-> E. x e. A ( f ` x ) = (/) ) )
3		raleq 2661	. . . . 5 |- ( y = A -> ( A. x e. y ( f ` x ) = 1o <-> A. x e. A ( f ` x ) = 1o ) )
4	2, 3	orbi12d 783	. . . 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 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 ) ) ) )
6	5	albidv 1812	. 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 7099	. 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 2873	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 104   \/ wo 698   A.wal 1341   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   (/)c0 3409   -->wf 5184   ` cfv 5188   1oc1o 6377   2oc2o 6378  Omnicomni 7098

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-fn 5191  df-f 5192  df-omni 7099

This theorem is referenced by:  isomnimap  7101  finomni  7104  exmidomniim  7105  exmidomni  7106  omnimkv  7120  omniwomnimkv  7131