Theorem isomni 7334

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 5466	. . . 4 |- ( y = A -> ( f : y --> 2o <-> f : A --> 2o ) )
2		rexeq 2731	. . . . 5 |- ( y = A -> ( E. x e. y ( f ` x ) = (/) <-> E. x e. A ( f ` x ) = (/) ) )
3		raleq 2730	. . . . 5 |- ( y = A -> ( A. x e. y ( f ` x ) = 1o <-> A. x e. A ( f ` x ) = 1o ) )
4	2, 3	orbi12d 800	. . . 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 1872	. 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 7333	. 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 2953	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 715   A.wal 1395   = wceq 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   (/)c0 3494   -->wf 5322   ` cfv 5326   1oc1o 6574   2oc2o 6575  Omnicomni 7332

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-fn 5329  df-f 5330  df-omni 7333

This theorem is referenced by:  isomnimap  7335  finomni  7338  exmidomniim  7339  exmidomni  7340  omnimkv  7354  omniwomnimkv  7365  nninfctlemfo  12610