Theorem fin23lem40 10287

Description: Lemma for fin23 10325. 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 8787	. . . 4 ⊢ (𝑓 ∈ (𝒫 𝐴 ↑m ω) → 𝑓:ω⟶𝒫 𝐴)
2		simpl 483	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → 𝐴 ∈ FinII)
3		frn 6675	. . . . . . 7 ⊢ (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ⊆ 𝒫 𝐴)
4	3	ad2antrl 726	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ran 𝑓 ⊆ 𝒫 𝐴)
5		fdm 6677	. . . . . . . . 9 ⊢ (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 = ω)
6		peano1 7825	. . . . . . . . . 10 ⊢ ∅ ∈ ω
7		ne0i 4294	. . . . . . . . . 10 ⊢ (∅ ∈ ω → ω ≠ ∅)
8	6, 7	mp1i 13	. . . . . . . . 9 ⊢ (𝑓:ω⟶𝒫 𝐴 → ω ≠ ∅)
9	5, 8	eqnetrd 3011	. . . . . . . 8 ⊢ (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 ≠ ∅)
10		dm0rn0 5880	. . . . . . . . 9 ⊢ (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
11	10	necon3bii 2996	. . . . . . . 8 ⊢ (dom 𝑓 ≠ ∅ ↔ ran 𝑓 ≠ ∅)
12	9, 11	sylib 217	. . . . . . 7 ⊢ (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ≠ ∅)
13	12	ad2antrl 726	. . . . . 6 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ran 𝑓 ≠ ∅)
14		ffn 6668	. . . . . . . . 9 ⊢ (𝑓:ω⟶𝒫 𝐴 → 𝑓 Fn ω)
15	14	ad2antrl 726	. . . . . . . 8 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → 𝑓 Fn ω)
16		sspss 4059	. . . . . . . . . . 11 ⊢ ((𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) ↔ ((𝑓‘suc 𝑏) ⊊ (𝑓‘𝑏) ∨ (𝑓‘suc 𝑏) = (𝑓‘𝑏)))
17		fvex 6855	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑏) ∈ V
18		fvex 6855	. . . . . . . . . . . . . 14 ⊢ (𝑓‘suc 𝑏) ∈ V
19	17, 18	brcnv 5838	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) [⊊] (𝑓‘𝑏))
20	17	brrpss 7663	. . . . . . . . . . . . 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 2743	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑏) = (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) = (𝑓‘𝑏))
23	21, 22	orbi12i 913	. . . . . . . . . . 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 3086	. . . . . . . . 9 ⊢ (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∀𝑏 ∈ ω ((𝑓‘𝑏)◡ [⊊] (𝑓‘suc 𝑏) ∨ (𝑓‘𝑏) = (𝑓‘suc 𝑏)))
26	25	ad2antll 727	. . . . . . . 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 7664	. . . . . . . . . 10 ⊢ [⊊] Po ran 𝑓
28		cnvpo 6239	. . . . . . . . . 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 10213	. . . . . . . 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 1371	. . . . . . 7 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ◡ [⊊] Or ran 𝑓)
33		cnvso 6240	. . . . . . 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 10254	. . . . . 6 ⊢ (((𝐴 ∈ FinII ∧ ran 𝑓 ⊆ 𝒫 𝐴) ∧ (ran 𝑓 ≠ ∅ ∧ [⊊] Or ran 𝑓)) → ∩ ran 𝑓 ∈ ran 𝑓)
36	2, 4, 13, 34, 35	syl22anc 837	. . . . 5 ⊢ ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏))) → ∩ ran 𝑓 ∈ ran 𝑓)
37	36	expr 457	. . . 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 3143	. 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 10265	. 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 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  {cab 2713   ≠ wne 2943  ∀wral 3064   ⊆ wss 3910   ⊊ wpss 3911  ∅c0 4282  𝒫 cpw 4560  ∩ cint 4907   class class class wbr 5105   Po wpo 5543   Or wor 5544  ^◡ccnv 5632  dom cdm 5633  ran crn 5634  suc csuc 6319   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   [⊊] crpss 7659  ωcom 7802   ↑_m cmap 8765  Fin^IIcfin2 10215

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-rpss 7660  df-om 7803  df-1st 7921  df-2nd 7922  df-map 8767  df-fin2 10222

This theorem is referenced by:  fin23  10325