Theorem isomnimap 7379

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 7378	. . 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 6732	. . . . . 6 |- 2o e. om
3		elmapg 6873	. . . . . 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 1872	. . 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 2516	. 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 716   A.wal 1396   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   (/)c0 3496   omcom 4694   -->wf 5329   ` cfv 5333  (class class class)co 6028   1oc1o 6618   2oc2o 6619   ^m cmap 6860  Omnicomni 7376

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-id 4396  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1o 6625  df-2o 6626  df-map 6862  df-omni 7377

This theorem is referenced by:  enomnilem  7380  fodjuomnilemres  7390  nninfomnilem  16727  isomninnlem  16745