Theorem fin23lem40 10442

Description: Lemma for fin23 10480. 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 8883	. . . 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 6728	. . . . . . 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 6730	. . . . . . . . 9 ⊢ (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 = ω)
6		peano1 7910	. . . . . . . . . 10 ⊢ ∅ ∈ ω
7		ne0i 4290	. . . . . . . . . 10 ⊢ (∅ ∈ ω → ω ≠ ∅)
8	6, 7	mp1i 14	. . . . . . . . 9 ⊢ (𝑓:ω⟶𝒫 𝐴 → ω ≠ ∅)
9	5, 8	eqnetrd 3026	. . . . . . . 8 ⊢ (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 ≠ ∅)
10		dm0rn0 5911	. . . . . . . . 9 ⊢ (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
11	10	necon3bii 3011	. . . . . . . 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 6720	. . . . . . . . 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 4032	. . . . . . . . . . 11 ⊢ ((𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) ↔ ((𝑓‘suc 𝑏) ⊊ (𝑓‘𝑏) ∨ (𝑓‘suc 𝑏) = (𝑓‘𝑏)))
17		fvex 6909	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑏) ∈ V
18		fvex 6909	. . . . . . . . . . . . . 14 ⊢ (𝑓‘suc 𝑏) ∈ V
19	17, 18	brcnv 5865	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) [⊊] (𝑓‘𝑏))
20	17	brrpss 7752	. . . . . . . . . . . . 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 2770	. . . . . . . . . . . 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 3103	. . . . . . . . 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 7753	. . . . . . . . . 10 ⊢ [⊊] Po ran 𝑓
28		cnvpo 6297	. . . . . . . . . 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 10368	. . . . . . . 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 6298	. . . . . . 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 10409	. . . . . 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 3158	. 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 10420	. 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 1569   ∈ wcel 2146  {cab 2741   ≠ wne 2959  ∀wral 3080   ⊆ wss 3900   ⊊ wpss 4016  ∅c0 4281  𝒫 cpw 4560  ∩ cint 4910   class class class wbr 5106   Po wpo 5561   Or wor 5562  ^◡ccnv 5654  dom cdm 5655  ran crn 5656  suc csuc 6371   Fn wfn 6542  ⟶wf 6543  ‘cfv 6547  (class class class)co 7430   [⊊] crpss 7748  ωcom 7887   ↑_m cmap 8861  Fin^IIcfin2 10370

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  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 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-rpss 7749  df-om 7888  df-1st 8011  df-2nd 8012  df-map 8863  df-fin2 10377

This theorem is used by:  fin23  10480