Theorem isomni 7080

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 5304	. . . 4 |- ( y = A -> ( f : y --> 2o <-> f : A --> 2o ) )
2		rexeq 2653	. . . . 5 |- ( y = A -> ( E. x e. y ( f ` x ) = (/) <-> E. x e. A ( f ` x ) = (/) ) )
3		raleq 2652	. . . . 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 1804	. 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 7079	. 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 2859	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 1333   = wceq 1335   e. wcel 2128   A.wral 2435   E.wrex 2436   (/)c0 3394   -->wf 5167   ` cfv 5171   1oc1o 6357   2oc2o 6358  Omnicomni 7078

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-fn 5174  df-f 5175  df-omni 7079

This theorem is referenced by:  isomnimap  7081  finomni  7084  exmidomniim  7085  exmidomni  7086  omnimkv  7100  omniwomnimkv  7111