Theorem iswomni 7499

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 5515	. . . 4 |- ( y = A -> ( f : y --> 2o <-> f : A --> 2o ) )
2		raleq 2749	. . . . 5 |- ( y = A -> ( A. x e. y ( f ` x ) = 1o <-> A. x e. A ( f ` x ) = 1o ) )
3	2	dcbid 850	. . . 4 |- ( y = A -> (DECID A. x e. y ( f ` x ) = 1o <-> DECID A. x e. A ( f ` x ) = 1o ) )
4	1, 3	imbi12d 234	. . 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 1877	. 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 7498	. 2 |- WOmni = { y | A. f ( f : y --> 2o -> DECID A. x e. y ( f ` x ) = 1o ) }
7	5, 6	elab2g 2973	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 105  DECID wdc 846   A.wal 1400   = wceq 1402   e. wcel 2209   A.wral 2528   -->wf 5371   ` cfv 5375   1oc1o 6674   2oc2o 6675  WOmnicwomni 7497

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-fn 5378  df-f 5379  df-womni 7498

This theorem is referenced by:  iswomnimap  7500  omniwomnimkv  7501