Theorem fin23lem41 10269

Description: Lemma for fin23 10306. 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 8900	. . . . 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 10262	. . . . . . . . 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 483	. . . . . . . 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 6732	. . . . . . . . . . 11 ⊢ ((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝒫 𝐴 ⊆ V) → 𝑏:ω–1-1→V)
7	5, 6	mpan2 698	. . . . . . . . . 10 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → 𝑏:ω–1-1→V)
8	7	ad2antrr 733	. . . . . . . . 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 6727	. . . . . . . . . . . 12 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → 𝑏:ω⟶𝒫 𝐴)
10		frn 6666	. . . . . . . . . . . 12 ⊢ (𝑏:ω⟶𝒫 𝐴 → ran 𝑏 ⊆ 𝒫 𝐴)
11		uniss 4849	. . . . . . . . . . . 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 5392	. . . . . . . . . . 11 ⊢ ∪ 𝒫 𝐴 = 𝐴
14	12, 13	sseqtrdi 3957	. . . . . . . . . 10 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → ∪ ran 𝑏 ⊆ 𝐴)
15	14	ad2antrr 733	. . . . . . . . 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 6722	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑒 → (𝑑:ω–1-1→V ↔ 𝑒:ω–1-1→V))
17		rneq 5885	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → ran 𝑑 = ran 𝑒)
18	17	unieqd 4854	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → ∪ ran 𝑑 = ∪ ran 𝑒)
19	18	sseq1d 3948	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑒 → (∪ ran 𝑑 ⊆ 𝐴 ↔ ∪ ran 𝑒 ⊆ 𝐴))
20	16, 19	anbi12d 639	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑒 → ((𝑑:ω–1-1→V ∧ ∪ ran 𝑑 ⊆ 𝐴) ↔ (𝑒:ω–1-1→V ∧ ∪ ran 𝑒 ⊆ 𝐴)))
21		fveq2 6831	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → (𝑐‘𝑑) = (𝑐‘𝑒))
22		f1eq1 6722	. . . . . . . . . . . . . 14 ⊢ ((𝑐‘𝑑) = (𝑐‘𝑒) → ((𝑐‘𝑑):ω–1-1→V ↔ (𝑐‘𝑒):ω–1-1→V))
23	21, 22	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑒 → ((𝑐‘𝑑):ω–1-1→V ↔ (𝑐‘𝑒):ω–1-1→V))
24	21	rneqd 5887	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑒 → ran (𝑐‘𝑑) = ran (𝑐‘𝑒))
25	24	unieqd 4854	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑒 → ∪ ran (𝑐‘𝑑) = ∪ ran (𝑐‘𝑒))
26	25, 18	psseq12d 4031	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑒 → (∪ ran (𝑐‘𝑑) ⊊ ∪ ran 𝑑 ↔ ∪ ran (𝑐‘𝑒) ⊊ ∪ ran 𝑒))
27	23, 26	anbi12d 639	. . . . . . . . . . . 12 ⊢ (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 346	. . . . . . . . . . 11 ⊢ (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 2044	. . . . . . . . . 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)))
30	29	bilani 506	. . . . . . . . 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)))
31		eqid 2741	. . . . . . . . 9 ⊢ (rec(𝑐, 𝑏) ↾ ω) = (rec(𝑐, 𝑏) ↾ ω)
32	2, 8, 15, 30, 31	fin23lem39 10267	. . . . . . . 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)
33	4, 32	exlimddv 1943	. . . . . . 7 ⊢ ((𝑏:ω–1-1→𝒫 𝐴 ∧ 𝐴 ∈ 𝐹) → ¬ 𝐴 ∈ 𝐹)
34	33	pm2.01da 805	. . . . . 6 ⊢ (𝑏:ω–1-1→𝒫 𝐴 → ¬ 𝐴 ∈ 𝐹)
35	34	exlimiv 1938	. . . . 5 ⊢ (∃𝑏 𝑏:ω–1-1→𝒫 𝐴 → ¬ 𝐴 ∈ 𝐹)
36	1, 35	syl 17	. . . 4 ⊢ (ω ≼ 𝒫 𝐴 → ¬ 𝐴 ∈ 𝐹)
37	36	con2i 139	. . 3 ⊢ (𝐴 ∈ 𝐹 → ¬ ω ≼ 𝒫 𝐴)
38		pwexg 5310	. . . 4 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ V)
39		isfin4-2 10231	. . . 4 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
40	38, 39	syl 17	. . 3 ⊢ (𝐴 ∈ 𝐹 → (𝒫 𝐴 ∈ FinIV ↔ ¬ ω ≼ 𝒫 𝐴))
41	37, 40	mpbird 259	. 2 ⊢ (𝐴 ∈ 𝐹 → 𝒫 𝐴 ∈ FinIV)
42		isfin3 10213	. 2 ⊢ (𝐴 ∈ FinIII ↔ 𝒫 𝐴 ∈ FinIV)
43	41, 42	sylibr 236	1 ⊢ (𝐴 ∈ 𝐹 → 𝐴 ∈ FinIII)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397  ∀wal 1546   = wceq 1548  ∃wex 1787   ∈ wcel 2121  {cab 2719  ∀wral 3055  Vcvv 3433   ⊆ wss 3885   ⊊ wpss 3886  𝒫 cpw 4532  ∪ cuni 4841  ∩ cint 4880   class class class wbr 5075  ran crn 5622   ↾ cres 5623  suc csuc 6316  ⟶wf 6485  –1-1→wf1 6486  ‘cfv 6489  (class class class)co 7360  ωcom 7810  reccrdg 8342   ↑_m cmap 8767   ≼ cdom 8885  Fin^IVcfin4 10197  Fin^IIIcfin3 10198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-seqom 8381  df-1o 8399  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-card 9858  df-fin4 10204  df-fin3 10205

This theorem is referenced by:  isf33lem  10283