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Theorem ssfin3ds 10371
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 5377 . . . 4 (𝐴𝐹 → 𝒫 𝐴 ∈ V)
2 simpr 484 . . . . 5 ((𝐴𝐹𝐵𝐴) → 𝐵𝐴)
32sspwd 4612 . . . 4 ((𝐴𝐹𝐵𝐴) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
4 mapss 8930 . . . 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 10370 . . . . 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 4051 . . 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 5322 . . . 4 ((𝐵𝐴𝐴𝐹) → 𝐵 ∈ V)
1312ancoms 458 . . 3 ((𝐴𝐹𝐵𝐴) → 𝐵 ∈ V)
146isfin3ds 10370 . . 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 1539  wcel 2107  {cab 2713  wral 3060  Vcvv 3479  wss 3950  𝒫 cpw 4599   cint 4945  ran crn 5685  suc csuc 6385  cfv 6560  (class class class)co 7432  ωcom 7888  m cmap 8867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-map 8869
This theorem is referenced by:  fin23lem31  10384
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