Theorem fin23lem40 10356

Description: Lemma for fin23 10394. Fin^II sets satisfy the descending chain condition. (Contributed by Stefan O'Rear, 3-Nov-2014.)

Hypothesis

Ref	Expression
fin23lem40.f	⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}

Assertion

Ref	Expression
fin23lem40	⊢ (𝐴 ∈ FinII → 𝐴 ∈ 𝐹)

Distinct variable groups:   𝑔,𝑎,𝑥,𝐴   𝐹,𝑎

Allowed substitution hints:   𝐹(𝑥, 𝑔)

Proof of Theorem fin23lem40

Dummy variables 𝑏 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapi 8851	. . . 4 ⊢ (𝑓 ∈ (𝒫 𝐴 ↑m ω) → 𝑓:ω⟶𝒫 𝐴)
2		simpl 488	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → 𝐴 ∈ FinII)
3		frn 6711	. . . . . . 7 ⊢ (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ⊆ 𝒫 𝐴)
4	3	ad2antrl 741	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ran 𝑓 ⊆ 𝒫 𝐴)
5		fdm 6713	. . . . . . . . 9 ⊢ (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 = ω)
6		peano1 7886	. . . . . . . . . 10 ⊢ ∅ ∈ ω
7		ne0i 4287	. . . . . . . . . 10 ⊢ (∅ ∈ ω → ω ≠ ∅)
8	6, 7	mp1i 14	. . . . . . . . 9 ⊢ (𝑓:ω⟶𝒫 𝐴 → ω ≠ ∅)
9	5, 8	eqnetrd 3022	. . . . . . . 8 ⊢ (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 ≠ ∅)
10		dm0rn0 5908	. . . . . . . . 9 ⊢ (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
11	10	necon3bii 3007	. . . . . . . 8 ⊢ (dom 𝑓 ≠ ∅ ↔ ran 𝑓 ≠ ∅)
12	9, 11	sylib 221	. . . . . . 7 ⊢ (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ≠ ∅)
13	12	ad2antrl 741	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ran 𝑓 ≠ ∅)
14		ffn 6703	. . . . . . . . 9 ⊢ (𝑓:ω⟶𝒫 𝐴 → 𝑓 Fn ω)
15	14	ad2antrl 741	. . . . . . . 8 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → 𝑓 Fn ω)
16		sspss 4050	. . . . . . . . . . 11 ⊢ ((𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) ↔ ((𝑓‘suc 𝑏) ⊊ (𝑓‘𝑏) ∨ (𝑓‘suc 𝑏) = (𝑓‘𝑏)))
17		fvex 6892	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑏) ∈ V
18		fvex 6892	. . . . . . . . . . . . . 14 ⊢ (𝑓‘suc 𝑏) ∈ V
19	17, 18	brcnv 5862	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) [⊊] (𝑓‘𝑏))
20	17	brrpss 7728	. . . . . . . . . . . . 13 ⊢ ((𝑓‘suc 𝑏) [⊊] (𝑓‘𝑏) ↔ (𝑓‘suc 𝑏) ⊊ (𝑓‘𝑏))
21	19, 20	bitri 278	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) ⊊ (𝑓‘𝑏))
22		eqcom 2767	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑏) = (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) = (𝑓‘𝑏))
23	21, 22	orbi12i 928	. . . . . . . . . . 11 ⊢ (((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ∨ (𝑓‘𝑏) = (𝑓‘suc 𝑏)) ↔ ((𝑓‘suc 𝑏) ⊊ (𝑓‘𝑏) ∨ (𝑓‘suc 𝑏) = (𝑓‘𝑏)))
24	16, 23	sylbb2 241	. . . . . . . . . 10 ⊢ ((𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ∨ (𝑓‘𝑏) = (𝑓‘suc 𝑏)))
25	24	ralimi 3099	. . . . . . . . 9 ⊢ (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∀𝑏 ∈ ω ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ∨ (𝑓‘𝑏) = (𝑓‘suc 𝑏)))
26	25	ad2antll 742	. . . . . . . 8 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ∀𝑏 ∈ ω ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ∨ (𝑓‘𝑏) = (𝑓‘suc 𝑏)))
27		porpss 7729	. . . . . . . . . 10 ⊢ [⊊] Po ran 𝑓
28		cnvpo 6285	. . . . . . . . . 10 ⊢ ( [⊊] Po ran 𝑓 ↔ ◡ [⊊] Po ran 𝑓)
29	27, 28	mpbi 233	. . . . . . . . 9 ⊢ ◡ [⊊] Po ran 𝑓
30	29	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ◡ [⊊] Po ran 𝑓)
31		sornom 10282	. . . . . . . 8 ⊢ ((𝑓 Fn ω ∧ ∀𝑏 ∈ ω ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ∨ (𝑓‘𝑏) = (𝑓‘suc 𝑏)) ∧ ◡ [⊊] Po ran 𝑓) → ◡ [⊊] Or ran 𝑓)
32	15, 26, 30, 31	syl3anc 1398	. . . . . . 7 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ◡ [⊊] Or ran 𝑓)
33		cnvso 6286	. . . . . . 7 ⊢ ( [⊊] Or ran 𝑓 ↔ ◡ [⊊] Or ran 𝑓)
34	32, 33	sylibr 237	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → [⊊] Or ran 𝑓)
35		fin2i2 10323	. . . . . 6 ⊢ (((𝐴 ∈ FinII ∧ ran 𝑓 ⊆ 𝒫 𝐴) ∧ (ran 𝑓 ≠ ∅ ∧ [⊊] Or ran 𝑓)) → ∩ ran 𝑓 ∈ ran 𝑓)
36	2, 4, 13, 34, 35	syl22anc 852	. . . . 5 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ∩ ran 𝑓 ∈ ran 𝑓)
37	36	expr 462	. . . 4 ⊢ ((𝐴 ∈ FinII ∧ 𝑓:ω⟶𝒫 𝐴) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∩ ran 𝑓 ∈ ran 𝑓))
38	1, 37	sylan2 605	. . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝑓 ∈ (𝒫 𝐴 ↑m ω)) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∩ ran 𝑓 ∈ ran 𝑓))
39	38	ralrimiva 3154	. 2 ⊢ (𝐴 ∈ FinII → ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∩ ran 𝑓 ∈ ran 𝑓))
40		fin23lem40.f	. . 3 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
41	40	isfin3ds 10334	. 2 ⊢ (𝐴 ∈ FinII → (𝐴 ∈ 𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∩ ran 𝑓 ∈ ran 𝑓)))
42	39, 41	mpbird 260	1 ⊢ (𝐴 ∈ FinII → 𝐴 ∈ 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  {cab 2738   ≠ wne 2955  ∀wral 3076   ⊆ wss 3899   ⊊ wpss 3900  ∅c0 4279  𝒫 cpw 4557  ∩ cint 4907   class class class wbr 5103   Po wpo 5561   Or wor 5562  ^◡ccnv 5654  dom cdm 5655  ran crn 5656  suc csuc 6359   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414   [⊊] crpss 7724  ωcom 7863   ↑_m cmap 8829  Fin^IIcfin2 10284

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-rpss 7725  df-om 7864  df-1st 7987  df-2nd 7988  df-map 8831  df-fin2 10291

This theorem is used by:  fin23  10394