Theorem iswomni 7166

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 5351	. . . 4 |- ( y = A -> ( f : y --> 2o <-> f : A --> 2o ) )
2		raleq 2673	. . . . 5 |- ( y = A -> ( A. x e. y ( f ` x ) = 1o <-> A. x e. A ( f ` x ) = 1o ) )
3	2	dcbid 838	. . . 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 1824	. 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 7165	. 2 |- WOmni = { y | A. f ( f : y --> 2o -> DECID A. x e. y ( f ` x ) = 1o ) }
7	5, 6	elab2g 2886	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 834   A.wal 1351   = wceq 1353   e. wcel 2148   A.wral 2455   -->wf 5214   ` cfv 5218   1oc1o 6413   2oc2o 6414  WOmnicwomni 7164

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-fn 5221  df-f 5222  df-womni 7165

This theorem is referenced by:  iswomnimap  7167  omniwomnimkv  7168