Theorem fin23lem40 10348

Description: Lemma for fin23 10386. 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 8845	. . . 4 ⊢ (𝑓 ∈ (𝒫 𝐴 ↑m ω) → 𝑓:ω⟶𝒫 𝐴)
2		simpl 481	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → 𝐴 ∈ FinII)
3		frn 6723	. . . . . . 7 ⊢ (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ⊆ 𝒫 𝐴)
4	3	ad2antrl 724	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ran 𝑓 ⊆ 𝒫 𝐴)
5		fdm 6725	. . . . . . . . 9 ⊢ (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 = ω)
6		peano1 7881	. . . . . . . . . 10 ⊢ ∅ ∈ ω
7		ne0i 4333	. . . . . . . . . 10 ⊢ (∅ ∈ ω → ω ≠ ∅)
8	6, 7	mp1i 13	. . . . . . . . 9 ⊢ (𝑓:ω⟶𝒫 𝐴 → ω ≠ ∅)
9	5, 8	eqnetrd 3006	. . . . . . . 8 ⊢ (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 ≠ ∅)
10		dm0rn0 5923	. . . . . . . . 9 ⊢ (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
11	10	necon3bii 2991	. . . . . . . 8 ⊢ (dom 𝑓 ≠ ∅ ↔ ran 𝑓 ≠ ∅)
12	9, 11	sylib 217	. . . . . . 7 ⊢ (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ≠ ∅)
13	12	ad2antrl 724	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ran 𝑓 ≠ ∅)
14		ffn 6716	. . . . . . . . 9 ⊢ (𝑓:ω⟶𝒫 𝐴 → 𝑓 Fn ω)
15	14	ad2antrl 724	. . . . . . . 8 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → 𝑓 Fn ω)
16		sspss 4098	. . . . . . . . . . 11 ⊢ ((𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) ↔ ((𝑓‘suc 𝑏) ⊊ (𝑓‘𝑏) ∨ (𝑓‘suc 𝑏) = (𝑓‘𝑏)))
17		fvex 6903	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑏) ∈ V
18		fvex 6903	. . . . . . . . . . . . . 14 ⊢ (𝑓‘suc 𝑏) ∈ V
19	17, 18	brcnv 5881	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) [⊊] (𝑓‘𝑏))
20	17	brrpss 7718	. . . . . . . . . . . . 13 ⊢ ((𝑓‘suc 𝑏) [⊊] (𝑓‘𝑏) ↔ (𝑓‘suc 𝑏) ⊊ (𝑓‘𝑏))
21	19, 20	bitri 274	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) ⊊ (𝑓‘𝑏))
22		eqcom 2737	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑏) = (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) = (𝑓‘𝑏))
23	21, 22	orbi12i 911	. . . . . . . . . . 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 237	. . . . . . . . . 10 ⊢ ((𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ∨ (𝑓‘𝑏) = (𝑓‘suc 𝑏)))
25	24	ralimi 3081	. . . . . . . . 9 ⊢ (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∀𝑏 ∈ ω ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ∨ (𝑓‘𝑏) = (𝑓‘suc 𝑏)))
26	25	ad2antll 725	. . . . . . . 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 7719	. . . . . . . . . 10 ⊢ [⊊] Po ran 𝑓
28		cnvpo 6285	. . . . . . . . . 10 ⊢ ( [⊊] Po ran 𝑓 ↔ ◡ [⊊] Po ran 𝑓)
29	27, 28	mpbi 229	. . . . . . . . 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 10274	. . . . . . . 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 1369	. . . . . . 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 233	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → [⊊] Or ran 𝑓)
35		fin2i2 10315	. . . . . 6 ⊢ (((𝐴 ∈ FinII ∧ ran 𝑓 ⊆ 𝒫 𝐴) ∧ (ran 𝑓 ≠ ∅ ∧ [⊊] Or ran 𝑓)) → ∩ ran 𝑓 ∈ ran 𝑓)
36	2, 4, 13, 34, 35	syl22anc 835	. . . . 5 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ∩ ran 𝑓 ∈ ran 𝑓)
37	36	expr 455	. . . 4 ⊢ ((𝐴 ∈ FinII ∧ 𝑓:ω⟶𝒫 𝐴) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∩ ran 𝑓 ∈ ran 𝑓))
38	1, 37	sylan2 591	. . 3 ⊢ ((𝐴 ∈ FinII ∧ 𝑓 ∈ (𝒫 𝐴 ↑m ω)) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∩ ran 𝑓 ∈ ran 𝑓))
39	38	ralrimiva 3144	. 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 10326	. 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 256	1 ⊢ (𝐴 ∈ FinII → 𝐴 ∈ 𝐹)

Syntax hints:   → wi 4   ∧ wa 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104  {cab 2707   ≠ wne 2938  ∀wral 3059   ⊆ wss 3947   ⊊ wpss 3948  ∅c0 4321  𝒫 cpw 4601  ∩ cint 4949   class class class wbr 5147   Po wpo 5585   Or wor 5586  ^◡ccnv 5674  dom cdm 5675  ran crn 5676  suc csuc 6365   Fn wfn 6537  ⟶wf 6538  ‘cfv 6542  (class class class)co 7411   [⊊] crpss 7714  ωcom 7857   ↑_m cmap 8822  Fin^IIcfin2 10276

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-rpss 7715  df-om 7858  df-1st 7977  df-2nd 7978  df-map 8824  df-fin2 10283

This theorem is referenced by:  fin23  10386