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Theorem ssfin3ds 9315
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 4987 . . . . 5 (𝐴𝐹 → 𝒫 𝐴 ∈ V)
21adantr 472 . . . 4 ((𝐴𝐹𝐵𝐴) → 𝒫 𝐴 ∈ V)
3 simpr 479 . . . . 5 ((𝐴𝐹𝐵𝐴) → 𝐵𝐴)
4 sspwb 5054 . . . . 5 (𝐵𝐴 ↔ 𝒫 𝐵 ⊆ 𝒫 𝐴)
53, 4sylib 208 . . . 4 ((𝐴𝐹𝐵𝐴) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
6 mapss 8054 . . . 4 ((𝒫 𝐴 ∈ V ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴) → (𝒫 𝐵𝑚 ω) ⊆ (𝒫 𝐴𝑚 ω))
72, 5, 6syl2anc 696 . . 3 ((𝐴𝐹𝐵𝐴) → (𝒫 𝐵𝑚 ω) ⊆ (𝒫 𝐴𝑚 ω))
8 isfin3ds.f . . . . . 6 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) → ran 𝑎 ∈ ran 𝑎)}
98isfin3ds 9314 . . . . 5 (𝐴𝐹 → (𝐴𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
109ibi 256 . . . 4 (𝐴𝐹 → ∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓))
1110adantr 472 . . 3 ((𝐴𝐹𝐵𝐴) → ∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓))
12 ssralv 3795 . . 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)))
137, 11, 12sylc 65 . 2 ((𝐴𝐹𝐵𝐴) → ∀𝑓 ∈ (𝒫 𝐵𝑚 ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓))
14 ssexg 4944 . . . 4 ((𝐵𝐴𝐴𝐹) → 𝐵 ∈ V)
1514ancoms 468 . . 3 ((𝐴𝐹𝐵𝐴) → 𝐵 ∈ V)
168isfin3ds 9314 . . 3 (𝐵 ∈ V → (𝐵𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐵𝑚 ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
1715, 16syl 17 . 2 ((𝐴𝐹𝐵𝐴) → (𝐵𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐵𝑚 ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
1813, 17mpbird 247 1 ((𝐴𝐹𝐵𝐴) → 𝐵𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1620  wcel 2127  {cab 2734  wral 3038  Vcvv 3328  wss 3703  𝒫 cpw 4290   cint 4615  ran crn 5255  suc csuc 5874  cfv 6037  (class class class)co 6801  ωcom 7218  𝑚 cmap 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-1st 7321  df-2nd 7322  df-map 8013
This theorem is referenced by:  fin23lem31  9328
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