Theorem fin23lem41 10039

Description: Lemma for fin23 10076. 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 8704	. . . . 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 10032	. . . . . . . . 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 3941	. . . . . . . . . . 11 ⊢ 𝒫 𝐴 ⊆ V
6		f1ss 6660	. . . . . . . . . . 11 ⊢ ((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝒫 𝐴 ⊆ V) → 𝑏:ω–1-1→V)
7	5, 6	mpan2 687	. . . . . . . . . 10 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → 𝑏:ω–1-1→V)
8	7	ad2antrr 722	. . . . . . . . 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 6654	. . . . . . . . . . . 12 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → 𝑏:ω⟶𝒫 𝐴)
10		frn 6591	. . . . . . . . . . . 12 ⊢ (𝑏:ω⟶𝒫 𝐴 → ran 𝑏 ⊆ 𝒫 𝐴)
11		uniss 4844	. . . . . . . . . . . 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 5360	. . . . . . . . . . 11 ⊢ ∪ 𝒫 𝐴 = 𝐴
14	12, 13	sseqtrdi 3967	. . . . . . . . . 10 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → ∪ ran 𝑏 ⊆ 𝐴)
15	14	ad2antrr 722	. . . . . . . . 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 6649	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (𝑑:ω–1-1→V ↔ 𝑒:ω–1-1→V))
17		rneq 5834	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑒 → ran 𝑑 = ran 𝑒)
18	17	unieqd 4850	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → ∪ ran 𝑑 = ∪ ran 𝑒)
19	18	sseq1d 3948	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (∪ ran 𝑑 ⊆ 𝐴 ↔ ∪ ran 𝑒 ⊆ 𝐴))
20	16, 19	anbi12d 630	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑒 → ((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) ↔ (𝑒:ω–1-1→V ∧ ∪ ran 𝑒 ⊆ 𝐴)))
21		fveq2 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → (𝑐‘𝑑) = (𝑐‘𝑒))
22		f1eq1 6649	. . . . . . . . . . . . . . 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 5836	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑒 → ran (𝑐‘𝑑) = ran (𝑐‘𝑒))
25	24	unieqd 4850	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → ∪ ran (𝑐‘𝑑) = ∪ ran (𝑐‘𝑒))
26	25, 18	psseq12d 4025	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑 ↔ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒))
27	23, 26	anbi12d 630	. . . . . . . . . . . . 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 2040	. . . . . . . . . . 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 215	. . . . . . . . . 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 2738	. . . . . . . . 9 ⊢ (rec(𝑐, 𝑏) ↾ ω) = (rec(𝑐, 𝑏) ↾ ω)
33	2, 8, 15, 31, 32	fin23lem39 10037	. . . . . . . 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 1939	. . . . . . 7 ⊢ ((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) → ¬ 𝐴 ∈ 𝐹)
35	34	pm2.01da 795	. . . . . 6 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → ¬ 𝐴 ∈ 𝐹)
36	35	exlimiv 1934	. . . . 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 5296	. . . 4 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ V)
40		isfin4-2 10001	. . . 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 256	. 2 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ FinIV)
43		isfin3 9983	. 2 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
44	42, 43	sylibr 233	1 ⊢ (𝐴 ∈ 𝐹 → 𝐴 ∈ FinIII)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715  ∀wral 3063  Vcvv 3422   ⊆ wss 3883   ⊊ wpss 3884  𝒫 cpw 4530  ∪ cuni 4836  ∩ cint 4876   class class class wbr 5070  ran crn 5581   ↾ cres 5582  suc csuc 6253  ⟶wf 6414  –1-1→wf1 6415  ‘cfv 6418  (class class class)co 7255  ωcom 7687  reccrdg 8211   ↑_m cmap 8573   ≼ cdom 8689  Fin^IVcfin4 9967  Fin^IIIcfin3 9968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-seqom 8249  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-fin4 9974  df-fin3 9975

This theorem is referenced by:  isf33lem  10053