Theorem ismkvmap 7447

Description: The predicate of being Markov stated in terms of set exponentiation. (Contributed by Jim Kingdon, 18-Mar-2023.)

Assertion

Ref	Expression
ismkvmap	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))

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

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem ismkvmap

Step	Hyp	Ref	Expression
1		ismkv 7446	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
2		2onn 6756	. . . . . 6 ⊢ 2o ∈ ω
3		elmapg 6897	. . . . . 6 ⊢ ((2o ∈ ω ∧ 𝐴 ∈ 𝑉) → (𝑓 ∈ (2o ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶2o))
4	2, 3	mpan 424	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑓 ∈ (2o ↑𝑚 𝐴) ↔ 𝑓:𝐴⟶2o))
5	4	imbi1d 231	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑓 ∈ (2o ↑𝑚 𝐴) → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)) ↔ (𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
6	5	albidv 1873	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑓(𝑓 ∈ (2o ↑𝑚 𝐴) → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)) ↔ ∀𝑓(𝑓:𝐴⟶2o → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
7	1, 6	bitr4d 191	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓(𝑓 ∈ (2o ↑𝑚 𝐴) → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅))))
8		df-ral 2527	. 2 ⊢ (∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅) ↔ ∀𝑓(𝑓 ∈ (2o ↑𝑚 𝐴) → (¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))
9	7, 8	bitr4di 198	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Markov ↔ ∀𝑓 ∈ (2o ↑𝑚 𝐴)(¬ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 1o → ∃𝑥 ∈ 𝐴 (𝑓‘𝑥) = ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105  ∀wal 1396   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  ∅c0 3510  ωcom 4714  ⟶wf 5350  ‘cfv 5354  (class class class)co 6052  1_oc1o 6642  2_oc2o 6643   ↑_𝑚 cmap 6884  Markovcmarkov 7444

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-id 4416  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1o 6649  df-2o 6650  df-map 6886  df-markov 7445

This theorem is referenced by:  ismkvnex  7448  fodjumkvlemres  7452  enmkvlem  7454  ismkvnnlem  16886