Theorem ssfin3ds 10283

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 5333	. . . 4 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ V)
2		simpr 484	. . . . 5 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
3	2	sspwd 4576	. . . 4 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
4		mapss 8862	. . . 4 ⊢ ((𝒫 𝐴 ∈ V ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴) → (𝒫 𝐵 ↑m ω) ⊆ (𝒫 𝐴 ↑m ω))
5	1, 3, 4	syl2an2r 685	. . 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 10282	. . . . 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 4015	. . 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 5278	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐹) → 𝐵 ∈ V)
13	12	ancoms 458	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
14	6	isfin3ds 10282	. . 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  Vcvv 3447   ⊆ wss 3914  𝒫 cpw 4563  ∩ cint 4910  ran crn 5639  suc csuc 6334  ‘cfv 6511  (class class class)co 7387  ωcom 7842   ↑_m cmap 8799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-map 8801

This theorem is referenced by:  fin23lem31  10296