Theorem isfin3ds 10326

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 6429	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → suc 𝑏 = suc 𝑥)
2	1	fveq2d 6894	. . . . . . . 8 ⊢ (𝑏 = 𝑥 → (𝑎‘suc 𝑏) = (𝑎‘suc 𝑥))
3		fveq2 6890	. . . . . . . 8 ⊢ (𝑏 = 𝑥 → (𝑎‘𝑏) = (𝑎‘𝑥))
4	2, 3	sseq12d 4014	. . . . . . 7 ⊢ (𝑏 = 𝑥 → ((𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) ↔ (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥)))
5	4	cbvralvw 3232	. . . . . 6 ⊢ (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) ↔ ∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥))
6		fveq1 6889	. . . . . . . 8 ⊢ (𝑎 = 𝑓 → (𝑎‘suc 𝑥) = (𝑓‘suc 𝑥))
7		fveq1 6889	. . . . . . . 8 ⊢ (𝑎 = 𝑓 → (𝑎‘𝑥) = (𝑓‘𝑥))
8	6, 7	sseq12d 4014	. . . . . . 7 ⊢ (𝑎 = 𝑓 → ((𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥)))
9	8	ralbidv 3175	. . . . . 6 ⊢ (𝑎 = 𝑓 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥)))
10	5, 9	bitrid 282	. . . . 5 ⊢ (𝑎 = 𝑓 → (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥)))
11		rneq 5934	. . . . . . 7 ⊢ (𝑎 = 𝑓 → ran 𝑎 = ran 𝑓)
12	11	inteqd 4954	. . . . . 6 ⊢ (𝑎 = 𝑓 → ∩ ran 𝑎 = ∩ ran 𝑓)
13	12, 11	eleq12d 2825	. . . . 5 ⊢ (𝑎 = 𝑓 → (∩ ran 𝑎 ∈ ran 𝑎 ↔ ∩ ran 𝑓 ∈ ran 𝑓))
14	10, 13	imbi12d 343	. . . 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 3232	. . 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 4615	. . . . 5 ⊢ (𝑔 = 𝐴 → 𝒫 𝑔 = 𝒫 𝐴)
17	16	oveq1d 7426	. . . 4 ⊢ (𝑔 = 𝐴 → (𝒫 𝑔 ↑m ω) = (𝒫 𝐴 ↑m ω))
18	17	raleqdv 3323	. . 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	bitrid 282	. 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 3669	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 205   = wceq 1539   ∈ wcel 2104  {cab 2707  ∀wral 3059   ⊆ wss 3947  𝒫 cpw 4601  ∩ cint 4949  ran crn 5676  suc csuc 6365  ‘cfv 6542  (class class class)co 7411  ωcom 7857   ↑_m cmap 8822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-cnv 5683  df-dm 5685  df-rn 5686  df-suc 6369  df-iota 6494  df-fv 6550  df-ov 7414

This theorem is referenced by:  ssfin3ds  10327  fin23lem17  10335  fin23lem39  10347  fin23lem40  10348  isf32lem12  10361  isfin3-3  10365