Theorem iswomni 7141

Description: The predicate of being weakly omniscient. (Contributed by Jim Kingdon, 9-Jun-2024.)

Assertion

Ref	Expression
iswomni	|- ( A e. V -> ( A e. WOmni <-> A. f ( f : A --> 2o -> DECID A. x e. A ( f ` x ) = 1o ) ) )

Distinct variable group:   A, f, x

Allowed substitution hints:   V( x, f)

Proof of Theorem iswomni

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		raleq 2665	. . . . 5 |- ( y = A -> ( A. x e. y ( f ` x ) = 1o <-> A. x e. A ( f ` x ) = 1o ) )
3	2	dcbid 833	. . . 4 |- ( y = A -> (DECID A. x e. y ( f ` x ) = 1o <-> DECID A. x e. A ( f ` x ) = 1o ) )
4	1, 3	imbi12d 233	. . 3 |- ( y = A -> ( ( f : y --> 2o -> DECID A. x e. y ( f ` x ) = 1o ) <-> ( f : A --> 2o -> DECID A. x e. A ( f ` x ) = 1o ) ) )
5	4	albidv 1817	. 2 |- ( y = A -> ( A. f ( f : y --> 2o -> DECID A. x e. y ( f ` x ) = 1o ) <-> A. f ( f : A --> 2o -> DECID A. x e. A ( f ` x ) = 1o ) ) )
6		df-womni 7140	. 2 |- WOmni = { y | A. f ( f : y --> 2o -> DECID A. x e. y ( f ` x ) = 1o ) }
7	5, 6	elab2g 2877	1 |- ( A e. V -> ( A e. WOmni <-> A. f ( f : A --> 2o -> DECID A. x e. A ( f ` x ) = 1o ) ) )

Syntax hints:   -> wi 4   <-> wb 104  DECID wdc 829   A.wal 1346   = wceq 1348   e. wcel 2141   A.wral 2448   -->wf 5194   ` cfv 5198   1oc1o 6388   2oc2o 6389  WOmnicwomni 7139

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-in1 609  ax-in2 610  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-dc 830  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-v 2732  df-fn 5201  df-f 5202  df-womni 7140

This theorem is referenced by:  iswomnimap  7142  omniwomnimkv  7143