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Theorem ssfin3ds 9803
 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 5251 . . . 4 (𝐴𝐹 → 𝒫 𝐴 ∈ V)
2 simpr 488 . . . . 5 ((𝐴𝐹𝐵𝐴) → 𝐵𝐴)
32sspwd 4512 . . . 4 ((𝐴𝐹𝐵𝐴) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
4 mapss 8484 . . . 4 ((𝒫 𝐴 ∈ V ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴) → (𝒫 𝐵m ω) ⊆ (𝒫 𝐴m ω))
51, 3, 4syl2an2r 684 . . 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 9802 . . . . 5 (𝐴𝐹 → (𝐴𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
87ibi 270 . . . 4 (𝐴𝐹 → ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓))
98adantr 484 . . 3 ((𝐴𝐹𝐵𝐴) → ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓))
10 ssralv 3960 . . 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 5197 . . . 4 ((𝐵𝐴𝐴𝐹) → 𝐵 ∈ V)
1312ancoms 462 . . 3 ((𝐴𝐹𝐵𝐴) → 𝐵 ∈ V)
146isfin3ds 9802 . . 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 260 1 ((𝐴𝐹𝐵𝐴) → 𝐵𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  Vcvv 3409   ⊆ wss 3860  𝒫 cpw 4497  ∩ cint 4841  ran crn 5529  suc csuc 6176  ‘cfv 6340  (class class class)co 7156  ωcom 7585   ↑m cmap 8422 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-map 8424 This theorem is referenced by:  fin23lem31  9816
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