Theorem isomnimap 7137

Description: The predicate of being omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 13-Jul-2022.)

Assertion

Ref	Expression
isomnimap	|- ( A e. V -> ( A e. Omni <-> A. f e. ( 2o ^m A ) ( E. x e. A ( f ` x ) = (/) \/ A. x e. A ( f ` x ) = 1o ) ) )

Distinct variable groups:   A, f, x   f, V

Allowed substitution hint:   V( x)

Proof of Theorem isomnimap

Step	Hyp	Ref	Expression
1		isomni 7136	. . 3 |- ( A e. V -> ( A e. Omni <-> A. f ( f : A --> 2o -> ( E. x e. A ( f ` x ) = (/) \/ A. x e. A ( f ` x ) = 1o ) ) ) )
2		2onn 6524	. . . . . 6 |- 2o e. om
3		elmapg 6663	. . . . . 6 |- ( ( 2o e. om /\ A e. V ) -> ( f e. ( 2o ^m A ) <-> f : A --> 2o ) )
4	2, 3	mpan 424	. . . . 5 |- ( A e. V -> ( f e. ( 2o ^m A ) <-> f : A --> 2o ) )
5	4	imbi1d 231	. . . 4 |- ( A e. V -> ( ( f e. ( 2o ^m A ) -> ( E. x e. A ( f ` x ) = (/) \/ A. x e. A ( f ` x ) = 1o ) ) <-> ( f : A --> 2o -> ( E. x e. A ( f ` x ) = (/) \/ A. x e. A ( f ` x ) = 1o ) ) ) )
6	5	albidv 1824	. . 3 |- ( A e. V -> ( A. f ( f e. ( 2o ^m A ) -> ( E. x e. A ( f ` x ) = (/) \/ A. x e. A ( f ` x ) = 1o ) ) <-> A. f ( f : A --> 2o -> ( E. x e. A ( f ` x ) = (/) \/ A. x e. A ( f ` x ) = 1o ) ) ) )
7	1, 6	bitr4d 191	. 2 |- ( A e. V -> ( A e. Omni <-> A. f ( f e. ( 2o ^m A ) -> ( E. x e. A ( f ` x ) = (/) \/ A. x e. A ( f ` x ) = 1o ) ) ) )
8		df-ral 2460	. 2 |- ( A. f e. ( 2o ^m A ) ( E. x e. A ( f ` x ) = (/) \/ A. x e. A ( f ` x ) = 1o ) <-> A. f ( f e. ( 2o ^m A ) -> ( E. x e. A ( f ` x ) = (/) \/ A. x e. A ( f ` x ) = 1o ) ) )
9	7, 8	bitr4di 198	1 |- ( A e. V -> ( A e. Omni <-> A. f e. ( 2o ^m A ) ( E. x e. A ( f ` x ) = (/) \/ A. x e. A ( f ` x ) = 1o ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 708   A.wal 1351   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   (/)c0 3424   omcom 4591   -->wf 5214   ` cfv 5218  (class class class)co 5877   1oc1o 6412   2oc2o 6413   ^m cmap 6650  Omnicomni 7134

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-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1o 6419  df-2o 6420  df-map 6652  df-omni 7135

This theorem is referenced by:  enomnilem  7138  fodjuomnilemres  7148  nninfomnilem  14852  isomninnlem  14863