Theorem ssfin3ds 10314

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 𝑔 ↑m ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) → ∩ ran 𝑎 ∈ ran 𝑎)}

Assertion

Ref	Expression
ssfin3ds	⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝐹)

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.

Step	Hyp	Ref	Expression
1		pwexg 5350	. . . 4 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ V)
2		simpr 489	. . . . 5 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
3	2	sspwd 4580	. . . 4 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
4		mapss 8887	. . . 4 ⊢ ((𝒫 𝐴 ∈ V ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴) → (𝒫 𝐵 ↑m ω) ⊆ (𝒫 𝐴 ↑m ω))
5	1, 3, 4	syl2an2r 697	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → (𝒫 𝐵 ↑m ω) ⊆ (𝒫 𝐴 ↑m ω))
6		isfin3ds.f	. . . . . 6 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) → ∩ ran 𝑎 ∈ ran 𝑎)}
7	6	isfin3ds 10313	. . . . 5 ⊢ (𝐴 ∈ 𝐹 → (𝐴 ∈ 𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓)))
8	7	ibi 270	. . . 4 ⊢ (𝐴 ∈ 𝐹 → ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓))
9	8	adantr 485	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓))
10		ssralv 4014	. . 3 ⊢ ((𝒫 𝐵 ↑m ω) ⊆ (𝒫 𝐴 ↑m ω) → (∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓) → ∀𝑓 ∈ (𝒫 𝐵 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓)))
11	5, 9, 10	sylc 66	. 2 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → ∀𝑓 ∈ (𝒫 𝐵 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓))
12		ssexg 5294	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐹) → 𝐵 ∈ V)
13	12	ancoms 463	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
14	6	isfin3ds 10313	. . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ 𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐵 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓)))
15	13, 14	syl 18	. 2 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∈ 𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐵 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓)))
16	11, 15	mpbird 260	1 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝐹)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  Vcvv 3463   ⊆ wss 3913  𝒫 cpw 4567  ∩ cint 4916  ran crn 5663  suc csuc 6363  ‘cfv 6537  (class class class)co 7411  ωcom 7862   ↑_m cmap 8824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-map 8826

This theorem is referenced by:  fin23lem31  10327