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Theorem iswomnimap 7211
Description: The predicate of being weakly omniscient stated in terms of set exponentiation. (Contributed by Jim Kingdon, 9-Jun-2024.)
Assertion
Ref Expression
iswomnimap  |-  ( A  e.  V  ->  ( A  e. WOmni  <->  A. f  e.  ( 2o  ^m  A )DECID  A. x  e.  A  (
f `  x )  =  1o ) )
Distinct variable groups:    A, f, x   
f, V
Allowed substitution hint:    V( x)

Proof of Theorem iswomnimap
StepHypRef Expression
1 iswomni 7210 . . 3  |-  ( A  e.  V  ->  ( A  e. WOmni  <->  A. f ( f : A --> 2o  -> DECID  A. x  e.  A  ( f `  x )  =  1o ) ) )
2 2onn 6561 . . . . . 6  |-  2o  e.  om
3 elmapg 6702 . . . . . 6  |-  ( ( 2o  e.  om  /\  A  e.  V )  ->  ( f  e.  ( 2o  ^m  A )  <-> 
f : A --> 2o ) )
42, 3mpan 424 . . . . 5  |-  ( A  e.  V  ->  (
f  e.  ( 2o 
^m  A )  <->  f : A
--> 2o ) )
54imbi1d 231 . . . 4  |-  ( A  e.  V  ->  (
( f  e.  ( 2o  ^m  A )  -> DECID  A. x  e.  A  ( f `  x
)  =  1o )  <-> 
( f : A --> 2o  -> DECID  A. x  e.  A  ( f `  x
)  =  1o ) ) )
65albidv 1835 . . 3  |-  ( A  e.  V  ->  ( A. f ( f  e.  ( 2o  ^m  A
)  -> DECID  A. x  e.  A  ( f `  x
)  =  1o )  <->  A. f ( f : A --> 2o  -> DECID  A. x  e.  A  ( f `  x
)  =  1o ) ) )
71, 6bitr4d 191 . 2  |-  ( A  e.  V  ->  ( A  e. WOmni  <->  A. f ( f  e.  ( 2o  ^m  A )  -> DECID  A. x  e.  A  ( f `  x
)  =  1o ) ) )
8 df-ral 2473 . 2  |-  ( A. f  e.  ( 2o  ^m  A )DECID 
A. x  e.  A  ( f `  x
)  =  1o  <->  A. f
( f  e.  ( 2o  ^m  A )  -> DECID  A. x  e.  A  ( f `  x
)  =  1o ) )
97, 8bitr4di 198 1  |-  ( A  e.  V  ->  ( A  e. WOmni  <->  A. f  e.  ( 2o  ^m  A )DECID  A. x  e.  A  (
f `  x )  =  1o ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105  DECID wdc 835   A.wal 1362    = wceq 1364    e. wcel 2160   A.wral 2468   omcom 4614   -->wf 5238   ` cfv 5242  (class class class)co 5906   1oc1o 6449   2oc2o 6450    ^m cmap 6689  WOmnicwomni 7208
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-nul 4151  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2758  df-sbc 2982  df-dif 3151  df-un 3153  df-in 3155  df-ss 3162  df-nul 3443  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-br 4026  df-opab 4087  df-id 4318  df-suc 4396  df-iom 4615  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-fv 5250  df-ov 5909  df-oprab 5910  df-mpo 5911  df-1o 6456  df-2o 6457  df-map 6691  df-womni 7209
This theorem is referenced by:  enwomnilem  7214  nninfdcinf  7216  nninfwlporlem  7218  nninfwlpoim  7223  iswomninnlem  15463
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