Theorem fin23lem41 9763

Description: Lemma for fin23 9800. 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 8503	. . . . 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 9756	. . . . . . . . 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 485	. . . . . . . 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 3939	. . . . . . . . . . 11 ⊢ 𝒫 𝐴 ⊆ V
6		f1ss 6555	. . . . . . . . . . 11 ⊢ ((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝒫 𝐴 ⊆ V) → 𝑏:ω–1-1→V)
7	5, 6	mpan2 690	. . . . . . . . . 10 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → 𝑏:ω–1-1→V)
8	7	ad2antrr 725	. . . . . . . . 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 6549	. . . . . . . . . . . 12 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → 𝑏:ω⟶𝒫 𝐴)
10		frn 6493	. . . . . . . . . . . 12 ⊢ (𝑏:ω⟶𝒫 𝐴 → ran 𝑏 ⊆ 𝒫 𝐴)
11		uniss 4808	. . . . . . . . . . . 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 5308	. . . . . . . . . . 11 ⊢ ∪ 𝒫 𝐴 = 𝐴
14	12, 13	sseqtrdi 3965	. . . . . . . . . 10 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → ∪ ran 𝑏 ⊆ 𝐴)
15	14	ad2antrr 725	. . . . . . . . 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 6544	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (𝑑:ω–1-1→V ↔ 𝑒:ω–1-1→V))
17		rneq 5770	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑒 → ran 𝑑 = ran 𝑒)
18	17	unieqd 4814	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → ∪ ran 𝑑 = ∪ ran 𝑒)
19	18	sseq1d 3946	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (∪ ran 𝑑 ⊆ 𝐴 ↔ ∪ ran 𝑒 ⊆ 𝐴))
20	16, 19	anbi12d 633	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑒 → ((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) ↔ (𝑒:ω–1-1→V ∧ ∪ ran 𝑒 ⊆ 𝐴)))
21		fveq2 6645	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → (𝑐‘𝑑) = (𝑐‘𝑒))
22		f1eq1 6544	. . . . . . . . . . . . . . 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 5772	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑒 → ran (𝑐‘𝑑) = ran (𝑐‘𝑒))
25	24	unieqd 4814	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → ∪ ran (𝑐‘𝑑) = ∪ ran (𝑐‘𝑒))
26	25, 18	psseq12d 4022	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑 ↔ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒))
27	23, 26	anbi12d 633	. . . . . . . . . . . . 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 348	. . . . . . . . . . . 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 2043	. . . . . . . . . . 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 219	. . . . . . . . . 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 485	. . . . . . . . 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 2798	. . . . . . . . 9 ⊢ (rec(𝑐, 𝑏) ↾ ω) = (rec(𝑐, 𝑏) ↾ ω)
33	2, 8, 15, 31, 32	fin23lem39 9761	. . . . . . . 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 141	. . 3 ⊢ (𝐴 ∈ 𝐹 → ¬ ω ≼ 𝒫 𝐴)
39		pwexg 5244	. . . 4 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ V)
40		isfin4-2 9725	. . . 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 260	. 2 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ FinIV)
43		isfin3 9707	. 2 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
44	42, 43	sylibr 237	1 ⊢ (𝐴 ∈ 𝐹 → 𝐴 ∈ FinIII)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∀wral 3106  Vcvv 3441   ⊆ wss 3881   ⊊ wpss 3882  𝒫 cpw 4497  ∪ cuni 4800  ∩ cint 4838   class class class wbr 5030  ran crn 5520   ↾ cres 5521  suc csuc 6161  ⟶wf 6320  –1-1→wf1 6321  ‘cfv 6324  (class class class)co 7135  ωcom 7560  reccrdg 8028   ↑_m cmap 8389   ≼ cdom 8490  Fin^IVcfin4 9691  Fin^IIIcfin3 9692

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-seqom 8067  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-card 9352  df-fin4 9698  df-fin3 9699

This theorem is referenced by:  isf33lem  9777