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Theorem ssfin3ds 10321
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 5375 . . . 4 (𝐴𝐹 → 𝒫 𝐴 ∈ V)
2 simpr 485 . . . . 5 ((𝐴𝐹𝐵𝐴) → 𝐵𝐴)
32sspwd 4614 . . . 4 ((𝐴𝐹𝐵𝐴) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
4 mapss 8879 . . . 4 ((𝒫 𝐴 ∈ V ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴) → (𝒫 𝐵m ω) ⊆ (𝒫 𝐴m ω))
51, 3, 4syl2an2r 683 . . 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 10320 . . . . 5 (𝐴𝐹 → (𝐴𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
87ibi 266 . . . 4 (𝐴𝐹 → ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓))
98adantr 481 . . 3 ((𝐴𝐹𝐵𝐴) → ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓))
10 ssralv 4049 . . 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 459 . . 3 ((𝐴𝐹𝐵𝐴) → 𝐵 ∈ V)
146isfin3ds 10320 . . 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 256 1 ((𝐴𝐹𝐵𝐴) → 𝐵𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2709  wral 3061  Vcvv 3474  wss 3947  𝒫 cpw 4601   cint 4949  ran crn 5676  suc csuc 6363  cfv 6540  (class class class)co 7405  ωcom 7851  m cmap 8816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818
This theorem is referenced by:  fin23lem31  10334
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