Theorem fin23lem40 10245

Description: Lemma for fin23 10283. 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 8776	. . . 4 ⊢ (𝑓 ∈ (𝒫 𝐴 ↑m ω) → 𝑓:ω⟶𝒫 𝐴)
2		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → 𝐴 ∈ FinII)
3		frn 6659	. . . . . . 7 ⊢ (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ⊆ 𝒫 𝐴)
4	3	ad2antrl 728	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ran 𝑓 ⊆ 𝒫 𝐴)
5		fdm 6661	. . . . . . . . 9 ⊢ (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 = ω)
6		peano1 7822	. . . . . . . . . 10 ⊢ ∅ ∈ ω
7		ne0i 4292	. . . . . . . . . 10 ⊢ (∅ ∈ ω → ω ≠ ∅)
8	6, 7	mp1i 13	. . . . . . . . 9 ⊢ (𝑓:ω⟶𝒫 𝐴 → ω ≠ ∅)
9	5, 8	eqnetrd 2992	. . . . . . . 8 ⊢ (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 ≠ ∅)
10		dm0rn0 5867	. . . . . . . . 9 ⊢ (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
11	10	necon3bii 2977	. . . . . . . 8 ⊢ (dom 𝑓 ≠ ∅ ↔ ran 𝑓 ≠ ∅)
12	9, 11	sylib 218	. . . . . . 7 ⊢ (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ≠ ∅)
13	12	ad2antrl 728	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ran 𝑓 ≠ ∅)
14		ffn 6652	. . . . . . . . 9 ⊢ (𝑓:ω⟶𝒫 𝐴 → 𝑓 Fn ω)
15	14	ad2antrl 728	. . . . . . . 8 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → 𝑓 Fn ω)
16		sspss 4053	. . . . . . . . . . 11 ⊢ ((𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) ↔ ((𝑓‘suc 𝑏) ⊊ (𝑓‘𝑏) ∨ (𝑓‘suc 𝑏) = (𝑓‘𝑏)))
17		fvex 6835	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑏) ∈ V
18		fvex 6835	. . . . . . . . . . . . . 14 ⊢ (𝑓‘suc 𝑏) ∈ V
19	17, 18	brcnv 5825	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) [⊊] (𝑓‘𝑏))
20	17	brrpss 7662	. . . . . . . . . . . . 13 ⊢ ((𝑓‘suc 𝑏) [⊊] (𝑓‘𝑏) ↔ (𝑓‘suc 𝑏) ⊊ (𝑓‘𝑏))
21	19, 20	bitri 275	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) ⊊ (𝑓‘𝑏))
22		eqcom 2736	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑏) = (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) = (𝑓‘𝑏))
23	21, 22	orbi12i 914	. . . . . . . . . . 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 238	. . . . . . . . . 10 ⊢ ((𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ∨ (𝑓‘𝑏) = (𝑓‘suc 𝑏)))
25	24	ralimi 3066	. . . . . . . . 9 ⊢ (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∀𝑏 ∈ ω ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ∨ (𝑓‘𝑏) = (𝑓‘suc 𝑏)))
26	25	ad2antll 729	. . . . . . . 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 7663	. . . . . . . . . 10 ⊢ [⊊] Po ran 𝑓
28		cnvpo 6235	. . . . . . . . . 10 ⊢ ( [⊊] Po ran 𝑓 ↔ ◡ [⊊] Po ran 𝑓)
29	27, 28	mpbi 230	. . . . . . . . 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 10171	. . . . . . . 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 1373	. . . . . . 7 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ◡ [⊊] Or ran 𝑓)
33		cnvso 6236	. . . . . . 7 ⊢ ( [⊊] Or ran 𝑓 ↔ ◡ [⊊] Or ran 𝑓)
34	32, 33	sylibr 234	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → [⊊] Or ran 𝑓)
35		fin2i2 10212	. . . . . 6 ⊢ (((𝐴 ∈ FinII ∧ ran 𝑓 ⊆ 𝒫 𝐴) ∧ (ran 𝑓 ≠ ∅ ∧ [⊊] Or ran 𝑓)) → ∩ ran 𝑓 ∈ ran 𝑓)
36	2, 4, 13, 34, 35	syl22anc 838	. . . . 5 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ∩ ran 𝑓 ∈ ran 𝑓)
37	36	expr 456	. . . 4 ⊢ ((𝐴 ∈ FinII ∧ 𝑓:ω⟶𝒫 𝐴) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∩ ran 𝑓 ∈ ran 𝑓))
38	1, 37	sylan2 593	. . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝑓 ∈ (𝒫 𝐴 ↑m ω)) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∩ ran 𝑓 ∈ ran 𝑓))
39	38	ralrimiva 3121	. 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 10223	. 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 257	1 ⊢ (𝐴 ∈ FinII → 𝐴 ∈ 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  {cab 2707   ≠ wne 2925  ∀wral 3044   ⊆ wss 3903   ⊊ wpss 3904  ∅c0 4284  𝒫 cpw 4551  ∩ cint 4896   class class class wbr 5092   Po wpo 5525   Or wor 5526  ^◡ccnv 5618  dom cdm 5619  ran crn 5620  suc csuc 6309   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   [⊊] crpss 7658  ωcom 7799   ↑_m cmap 8753  Fin^IIcfin2 10173

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 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-rpss 7659  df-om 7800  df-1st 7924  df-2nd 7925  df-map 8755  df-fin2 10180

This theorem is referenced by:  fin23  10283