Theorem isomnimap 7130

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 7129	. . 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 6517	. . . . . 6 |- 2o e. om
3		elmapg 6656	. . . . . 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 3422   omcom 4587   -->wf 5209   ` cfv 5213  (class class class)co 5870   1oc1o 6405   2oc2o 6406   ^m cmap 6643  Omnicomni 7127

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 4119  ax-nul 4127  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534

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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-int 3844  df-br 4002  df-opab 4063  df-id 4291  df-suc 4369  df-iom 4588  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-fv 5221  df-ov 5873  df-oprab 5874  df-mpo 5875  df-1o 6412  df-2o 6413  df-map 6645  df-omni 7128

This theorem is referenced by:  enomnilem  7131  fodjuomnilemres  7141  nninfomnilem  14538  isomninnlem  14549