Theorem iswomnimap 7496

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 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))

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

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem iswomnimap

Step	Hyp	Ref	Expression
1		iswomni 7495	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
2		2onn 6784	. . . . . 6 ⊢ 2o ∈ ω
3		elmapg 6925	. . . . . 6 ⊢ ((2o ∈ ω ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (2o ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶2o))
4	2, 3	mpan 428	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (2o ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶2o))
5	4	imbi1d 231	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑓 ∈ (2o ↑𝑚 𝐴) → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) ↔ (𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
6	5	albidv 1877	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑓(𝑓 ∈ (2o ↑𝑚 𝐴) → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o) ↔ ∀𝑓(𝑓:𝐴⟶2o → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
7	1, 6	bitr4d 191	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓(𝑓 ∈ (2o ↑𝑚 𝐴) → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o)))
8		df-ral 2533	. 2 ⊢ (∀𝑓 ∈ (2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o ↔ ∀𝑓(𝑓 ∈ (2o ↑𝑚 𝐴) → DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))
9	7, 8	bitr4di 198	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)DECID ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o))

Syntax hints:   → wi 4   ↔ wb 105  DECID wdc 846  ∀wal 1400   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ωcom 4732  ⟶wf 5368  ‘cfv 5372  (class class class)co 6075  1_oc1o 6670  2_oc2o 6671   ↑_𝑚 cmap 6912  WOmnicwomni 7493

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1o 6677  df-2o 6678  df-map 6914  df-womni 7494

This theorem is referenced by:  enwomnilem  7499  nninfdcinf  7501  nninfwlporlem  7503  nninfwlpoim  7509  nninfinfwlpo  7510  iswomninnlem  17004