Theorem iswomni 7087

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 5296	. . . 4 |- ( y = A -> ( f : y --> 2o <-> f : A --> 2o ) )
2		raleq 2649	. . . . 5 |- ( y = A -> ( A. x e. y ( f ` x ) = 1o <-> A. x e. A ( f ` x ) = 1o ) )
3	2	dcbid 824	. . . 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 1801	. 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 7086	. 2 |- WOmni = { y | A. f ( f : y --> 2o -> DECID A. x e. y ( f ` x ) = 1o ) }
7	5, 6	elab2g 2855	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 820   A.wal 1330   = wceq 1332   e. wcel 2125   A.wral 2432   -->wf 5159   ` cfv 5163   1oc1o 6346   2oc2o 6347  WOmnicwomni 7085

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-v 2711  df-fn 5166  df-f 5167  df-womni 7086

This theorem is referenced by:  iswomnimap  7088  omniwomnimkv  7089