Theorem fin23lem41 10243

Description: Lemma for fin23 10280. A set which satisfies the descending sequence condition must be III-finite. (Contributed by Stefan O'Rear, 2-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
fin23lem41	⊢ (𝐴 ∈ 𝐹 → 𝐴 ∈ FinIII)

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

Allowed substitution hints:   𝐹(𝑥,𝑔)

Proof of Theorem fin23lem41

Dummy variables 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 8882	. . . . 5 ⊢ (ω ≼ 𝒫 𝐴 → ∃𝑏 𝑏:ω–1-1→𝒫 𝐴)
2		fin23lem40.f	. . . . . . . . . 10 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
3	2	fin23lem33 10236	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐹 → ∃𝑐∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑)))
4	3	adantl 481	. . . . . . . 8 ⊢ ((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) → ∃𝑐∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑)))
5		ssv 3954	. . . . . . . . . . 11 ⊢ 𝒫 𝐴 ⊆ V
6		f1ss 6724	. . . . . . . . . . 11 ⊢ ((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝒫 𝐴 ⊆ V) → 𝑏:ω–1-1→V)
7	5, 6	mpan2 691	. . . . . . . . . 10 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → 𝑏:ω–1-1→V)
8	7	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) ∧ ∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑))) → 𝑏:ω–1-1→V)
9		f1f 6719	. . . . . . . . . . . 12 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → 𝑏:ω⟶𝒫 𝐴)
10		frn 6658	. . . . . . . . . . . 12 ⊢ (𝑏:ω⟶𝒫 𝐴 → ran 𝑏 ⊆ 𝒫 𝐴)
11		uniss 4864	. . . . . . . . . . . 12 ⊢ (ran 𝑏 ⊆ 𝒫 𝐴 → ∪ ran 𝑏 ⊆ ∪ 𝒫 𝐴)
12	9, 10, 11	3syl 18	. . . . . . . . . . 11 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → ∪ ran 𝑏 ⊆ ∪ 𝒫 𝐴)
13		unipw 5389	. . . . . . . . . . 11 ⊢ ∪ 𝒫 𝐴 = 𝐴
14	12, 13	sseqtrdi 3970	. . . . . . . . . 10 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → ∪ ran 𝑏 ⊆ 𝐴)
15	14	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) ∧ ∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑))) → ∪ ran 𝑏 ⊆ 𝐴)
16		f1eq1 6714	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (𝑑:ω–1-1→V ↔ 𝑒:ω–1-1→V))
17		rneq 5875	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑒 → ran 𝑑 = ran 𝑒)
18	17	unieqd 4869	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → ∪ ran 𝑑 = ∪ ran 𝑒)
19	18	sseq1d 3961	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (∪ ran 𝑑 ⊆ 𝐴 ↔ ∪ ran 𝑒 ⊆ 𝐴))
20	16, 19	anbi12d 632	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑒 → ((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) ↔ (𝑒:ω–1-1→V ∧ ∪ ran 𝑒 ⊆ 𝐴)))
21		fveq2 6822	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → (𝑐‘𝑑) = (𝑐‘𝑒))
22		f1eq1 6714	. . . . . . . . . . . . . . 15 ⊢ ((𝑐‘𝑑) = (𝑐‘𝑒) → ((𝑐‘𝑑):ω–1-1→V ↔ (𝑐‘𝑒):ω–1-1→V))
23	21, 22	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → ((𝑐‘𝑑):ω–1-1→V ↔ (𝑐‘𝑒):ω–1-1→V))
24	21	rneqd 5877	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑒 → ran (𝑐‘𝑑) = ran (𝑐‘𝑒))
25	24	unieqd 4869	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → ∪ ran (𝑐‘𝑑) = ∪ ran (𝑐‘𝑒))
26	25, 18	psseq12d 4044	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑 ↔ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒))
27	23, 26	anbi12d 632	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑒 → (((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑) ↔ ((𝑐‘𝑒):ω–1-1→V ∧ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒)))
28	20, 27	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑒 → (((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑)) ↔ ((𝑒:ω–1-1→V ∧ ∪ ran 𝑒 ⊆ 𝐴) → ((𝑐‘𝑒):ω–1-1→V ∧ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒))))
29	28	cbvalvw 2037	. . . . . . . . . . 11 ⊢ (∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑)) ↔ ∀𝑒((𝑒:ω–1-1→V ∧ ∪ ran 𝑒 ⊆ 𝐴) → ((𝑐‘𝑒):ω–1-1→V ∧ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒)))
30	29	biimpi 216	. . . . . . . . . 10 ⊢ (∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑)) → ∀𝑒((𝑒:ω–1-1→V ∧ ∪ ran 𝑒 ⊆ 𝐴) → ((𝑐‘𝑒):ω–1-1→V ∧ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒)))
31	30	adantl 481	. . . . . . . . 9 ⊢ (((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) ∧ ∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑))) → ∀𝑒((𝑒:ω–1-1→V ∧ ∪ ran 𝑒 ⊆ 𝐴) → ((𝑐‘𝑒):ω–1-1→V ∧ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒)))
32		eqid 2731	. . . . . . . . 9 ⊢ (rec(𝑐, 𝑏) ↾ ω) = (rec(𝑐, 𝑏) ↾ ω)
33	2, 8, 15, 31, 32	fin23lem39 10241	. . . . . . . 8 ⊢ (((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) ∧ ∀𝑑((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) → ((𝑐‘𝑑):ω–1-1→V ∧ ∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑))) → ¬ 𝐴 ∈ 𝐹)
34	4, 33	exlimddv 1936	. . . . . . 7 ⊢ ((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) → ¬ 𝐴 ∈ 𝐹)
35	34	pm2.01da 798	. . . . . 6 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → ¬ 𝐴 ∈ 𝐹)
36	35	exlimiv 1931	. . . . 5 ⊢ (∃𝑏 𝑏:ω–1-1→𝒫 𝐴 → ¬ 𝐴 ∈ 𝐹)
37	1, 36	syl 17	. . . 4 ⊢ (ω ≼ 𝒫 𝐴 → ¬ 𝐴 ∈ 𝐹)
38	37	con2i 139	. . 3 ⊢ (𝐴 ∈ 𝐹 → ¬ ω ≼ 𝒫 𝐴)
39		pwexg 5314	. . . 4 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ V)
40		isfin4-2 10205	. . . 4 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
41	39, 40	syl 17	. . 3 ⊢ (𝐴 ∈ 𝐹 → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
42	38, 41	mpbird 257	. 2 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ FinIV)
43		isfin3 10187	. 2 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
44	42, 43	sylibr 234	1 ⊢ (𝐴 ∈ 𝐹 → 𝐴 ∈ FinIII)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∀wral 3047  Vcvv 3436   ⊆ wss 3897   ⊊ wpss 3898  𝒫 cpw 4547  ∪ cuni 4856  ∩ cint 4895   class class class wbr 5089  ran crn 5615   ↾ cres 5616  suc csuc 6308  ⟶wf 6477  –1-1→wf1 6478  ‘cfv 6481  (class class class)co 7346  ωcom 7796  reccrdg 8328   ↑_m cmap 8750   ≼ cdom 8867  Fin^IVcfin4 10171  Fin^IIIcfin3 10172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-seqom 8367  df-1o 8385  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-card 9832  df-fin4 10178  df-fin3 10179

This theorem is referenced by:  isf33lem  10257