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Theorem ssfin3ds 10368
Description: A subset of a III-finite set is III-finite. (Contributed by Stefan O'Rear, 4-Nov-2014.)
Hypothesis
Ref Expression
isfin3ds.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) → ran 𝑎 ∈ ran 𝑎)}
Assertion
Ref Expression
ssfin3ds ((𝐴𝐹𝐵𝐴) → 𝐵𝐹)
Distinct variable groups:   𝑎,𝑏,𝑔,𝐴   𝐵,𝑎,𝑏,𝑔
Allowed substitution hints:   𝐹(𝑔,𝑎,𝑏)

Proof of Theorem ssfin3ds
Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 5384 . . . 4 (𝐴𝐹 → 𝒫 𝐴 ∈ V)
2 simpr 484 . . . . 5 ((𝐴𝐹𝐵𝐴) → 𝐵𝐴)
32sspwd 4618 . . . 4 ((𝐴𝐹𝐵𝐴) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
4 mapss 8928 . . . 4 ((𝒫 𝐴 ∈ V ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴) → (𝒫 𝐵m ω) ⊆ (𝒫 𝐴m ω))
51, 3, 4syl2an2r 685 . . 3 ((𝐴𝐹𝐵𝐴) → (𝒫 𝐵m ω) ⊆ (𝒫 𝐴m ω))
6 isfin3ds.f . . . . . 6 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) → ran 𝑎 ∈ ran 𝑎)}
76isfin3ds 10367 . . . . 5 (𝐴𝐹 → (𝐴𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
87ibi 267 . . . 4 (𝐴𝐹 → ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓))
98adantr 480 . . 3 ((𝐴𝐹𝐵𝐴) → ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓))
10 ssralv 4064 . . 3 ((𝒫 𝐵m ω) ⊆ (𝒫 𝐴m ω) → (∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓) → ∀𝑓 ∈ (𝒫 𝐵m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
115, 9, 10sylc 65 . 2 ((𝐴𝐹𝐵𝐴) → ∀𝑓 ∈ (𝒫 𝐵m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓))
12 ssexg 5329 . . . 4 ((𝐵𝐴𝐴𝐹) → 𝐵 ∈ V)
1312ancoms 458 . . 3 ((𝐴𝐹𝐵𝐴) → 𝐵 ∈ V)
146isfin3ds 10367 . . 3 (𝐵 ∈ V → (𝐵𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐵m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
1513, 14syl 17 . 2 ((𝐴𝐹𝐵𝐴) → (𝐵𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐵m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
1611, 15mpbird 257 1 ((𝐴𝐹𝐵𝐴) → 𝐵𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  Vcvv 3478  wss 3963  𝒫 cpw 4605   cint 4951  ran crn 5690  suc csuc 6388  cfv 6563  (class class class)co 7431  ωcom 7887  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by:  fin23lem31  10381
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