Theorem isomni 7112

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 5331	. . . 4 |- ( y = A -> ( f : y --> 2o <-> f : A --> 2o ) )
2		rexeq 2666	. . . . 5 |- ( y = A -> ( E. x e. y ( f ` x ) = (/) <-> E. x e. A ( f ` x ) = (/) ) )
3		raleq 2665	. . . . 5 |- ( y = A -> ( A. x e. y ( f ` x ) = 1o <-> A. x e. A ( f ` x ) = 1o ) )
4	2, 3	orbi12d 788	. . . 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 1817	. 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 7111	. 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 2877	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 703   A.wal 1346   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   (/)c0 3414   -->wf 5194   ` cfv 5198   1oc1o 6388   2oc2o 6389  Omnicomni 7110

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-fn 5201  df-f 5202  df-omni 7111

This theorem is referenced by:  isomnimap  7113  finomni  7116  exmidomniim  7117  exmidomni  7118  omnimkv  7132  omniwomnimkv  7143