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 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 5251	. . . 4 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ V)
2		simpr 488	. . . . 5 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
3	2	sspwd 4512	. . . 4 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
4		mapss 8484	. . . 4 ⊢ ((𝒫 𝐴 ∈ V ∧ 𝒫 𝐵 ⊆ 𝒫 𝐴) → (𝒫 𝐵 ↑m ω) ⊆ (𝒫 𝐴 ↑m ω))
5	1, 3, 4	syl2an2r 684	. . 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 9802	. . . . 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 484	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓))
10		ssralv 3960	. . 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 5197	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐹) → 𝐵 ∈ V)
13	12	ancoms 462	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ V)
14	6	isfin3ds 9802	. . 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 260	1 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ 𝐹)

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