Theorem ssfin3ds 10331

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 5376	. . . 4 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ V)
2		simpr 484	. . . . 5 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
3	2	sspwd 4615	. . . 4 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
4		mapss 8889	. . . 4 ⊢ ((𝒫 𝐴 ∈ V ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴) → (𝒫 𝐵 ↑m ω) ⊆ (𝒫 𝐴 ↑m ω))
5	1, 3, 4	syl2an2r 682	. . 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 10330	. . . . 5 ⊢ (𝐴 ∈ 𝐹 → (𝐴 ∈ 𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓)))
8	7	ibi 267	. . . 4 ⊢ (𝐴 ∈ 𝐹 → ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓))
9	8	adantr 480	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓))
10		ssralv 4050	. . 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 65	. 2 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → ∀𝑓 ∈ (𝒫 𝐵 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓))
12		ssexg 5323	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐹) → 𝐵 ∈ V)
13	12	ancoms 458	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
14	6	isfin3ds 10330	. . 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 17	. 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 257	1 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {cab 2708  ∀wral 3060  Vcvv 3473   ⊆ wss 3948  𝒫 cpw 4602  ∩ cint 4950  ran crn 5677  suc csuc 6366  ‘cfv 6543  (class class class)co 7412  ωcom 7859   ↑_m cmap 8826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-map 8828

This theorem is referenced by:  fin23lem31  10344