Theorem isfin3ds 9740

Description: Property of a III-finite set (descending sequence version). (Contributed by Mario Carneiro, 16-May-2015.)

Hypothesis

Ref	Expression
isfin3ds.f	⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) → ∩ ran 𝑎 ∈ ran 𝑎)}

Assertion

Ref	Expression
isfin3ds	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓)))

Distinct variable group:   𝑎,𝑏,𝑓,𝑔,𝑥,𝐴

Allowed substitution hints:   𝐹(𝑥,𝑓,𝑔,𝑎,𝑏)   𝑉(𝑥,𝑓,𝑔,𝑎,𝑏)

Proof of Theorem isfin3ds

Step	Hyp	Ref	Expression
1		suceq 6224	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → suc 𝑏 = suc 𝑥)
2	1	fveq2d 6649	. . . . . . . 8 ⊢ (𝑏 = 𝑥 → (𝑎‘suc 𝑏) = (𝑎‘suc 𝑥))
3		fveq2 6645	. . . . . . . 8 ⊢ (𝑏 = 𝑥 → (𝑎‘𝑏) = (𝑎‘𝑥))
4	2, 3	sseq12d 3948	. . . . . . 7 ⊢ (𝑏 = 𝑥 → ((𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) ↔ (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥)))
5	4	cbvralvw 3396	. . . . . 6 ⊢ (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) ↔ ∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥))
6		fveq1 6644	. . . . . . . 8 ⊢ (𝑎 = 𝑓 → (𝑎‘suc 𝑥) = (𝑓‘suc 𝑥))
7		fveq1 6644	. . . . . . . 8 ⊢ (𝑎 = 𝑓 → (𝑎‘𝑥) = (𝑓‘𝑥))
8	6, 7	sseq12d 3948	. . . . . . 7 ⊢ (𝑎 = 𝑓 → ((𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥)))
9	8	ralbidv 3162	. . . . . 6 ⊢ (𝑎 = 𝑓 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥)))
10	5, 9	syl5bb 286	. . . . 5 ⊢ (𝑎 = 𝑓 → (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥)))
11		rneq 5770	. . . . . . 7 ⊢ (𝑎 = 𝑓 → ran 𝑎 = ran 𝑓)
12	11	inteqd 4843	. . . . . 6 ⊢ (𝑎 = 𝑓 → ∩ ran 𝑎 = ∩ ran 𝑓)
13	12, 11	eleq12d 2884	. . . . 5 ⊢ (𝑎 = 𝑓 → (∩ ran 𝑎 ∈ ran 𝑎 ↔ ∩ ran 𝑓 ∈ ran 𝑓))
14	10, 13	imbi12d 348	. . . 4 ⊢ (𝑎 = 𝑓 → ((∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) → ∩ ran 𝑎 ∈ ran 𝑎) ↔ (∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓)))
15	14	cbvralvw 3396	. . 3 ⊢ (∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) → ∩ ran 𝑎 ∈ ran 𝑎) ↔ ∀𝑓 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓))
16		pweq 4513	. . . . 5 ⊢ (𝑔 = 𝐴 → 𝒫 𝑔 = 𝒫 𝐴)
17	16	oveq1d 7150	. . . 4 ⊢ (𝑔 = 𝐴 → (𝒫 𝑔 ↑m ω) = (𝒫 𝐴 ↑m ω))
18	17	raleqdv 3364	. . 3 ⊢ (𝑔 = 𝐴 → (∀𝑓 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓) ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓)))
19	15, 18	syl5bb 286	. 2 ⊢ (𝑔 = 𝐴 → (∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) → ∩ ran 𝑎 ∈ ran 𝑎) ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓)))
20		isfin3ds.f	. 2 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) → ∩ ran 𝑎 ∈ ran 𝑎)}
21	19, 20	elab2g 3616	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴 ↑m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥) → ∩ ran 𝑓 ∈ ran 𝑓)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106   ⊆ wss 3881  𝒫 cpw 4497  ∩ cint 4838  ran crn 5520  suc csuc 6161  ‘cfv 6324  (class class class)co 7135  ωcom 7560   ↑_m cmap 8389

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

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-cnv 5527  df-dm 5529  df-rn 5530  df-suc 6165  df-iota 6283  df-fv 6332  df-ov 7138

This theorem is referenced by:  ssfin3ds  9741  fin23lem17  9749  fin23lem39  9761  fin23lem40  9762  isf32lem12  9775  isfin3-3  9779