Theorem isfin3ds 10288

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 6416	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → suc 𝑏 = suc 𝑥)
2	1	fveq2d 6873	. . . . . . . 8 ⊢ (𝑏 = 𝑥 → (𝑎‘suc 𝑏) = (𝑎‘suc 𝑥))
3		fveq2 6869	. . . . . . . 8 ⊢ (𝑏 = 𝑥 → (𝑎‘𝑏) = (𝑎‘𝑥))
4	2, 3	sseq12d 3971	. . . . . . 7 ⊢ (𝑏 = 𝑥 → ((𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) ↔ (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥)))
5	4	cbvralvw 3242	. . . . . 6 ⊢ (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) ↔ ∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥))
6		fveq1 6868	. . . . . . . 8 ⊢ (𝑎 = 𝑓 → (𝑎‘suc 𝑥) = (𝑓‘suc 𝑥))
7		fveq1 6868	. . . . . . . 8 ⊢ (𝑎 = 𝑓 → (𝑎‘𝑥) = (𝑓‘𝑥))
8	6, 7	sseq12d 3971	. . . . . . 7 ⊢ (𝑎 = 𝑓 → ((𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥)))
9	8	ralbidv 3187	. . . . . 6 ⊢ (𝑎 = 𝑓 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥)))
10	5, 9	bitrid 285	. . . . 5 ⊢ (𝑎 = 𝑓 → (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥)))
11		rneq 5914	. . . . . . 7 ⊢ (𝑎 = 𝑓 → ran 𝑎 = ran 𝑓)
12	11	inteqd 4912	. . . . . 6 ⊢ (𝑎 = 𝑓 → ∩ ran 𝑎 = ∩ ran 𝑓)
13	12, 11	eleq12d 2858	. . . . 5 ⊢ (𝑎 = 𝑓 → (∩ ran 𝑎 ∈ ran 𝑎 ↔ ∩ ran 𝑓 ∈ ran 𝑓))
14	10, 13	imbi12d 346	. . . 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 3242	. . 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 4571	. . . . 5 ⊢ (𝑔 = 𝐴 → 𝒫 𝑔 = 𝒫 𝐴)
17	16	oveq1d 7413	. . . 4 ⊢ (𝑔 = 𝐴 → (𝒫 𝑔 ↑m ω) = (𝒫 𝐴 ↑m ω))
18	17	raleqdv 3322	. . 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 285	. 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 3641	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 208   = wceq 1562   ∈ wcel 2144  {cab 2742  ∀wral 3078   ⊆ wss 3906  𝒫 cpw 4557  ∩ cint 4907  ran crn 5650  suc csuc 6350  ‘cfv 6523  (class class class)co 7398  ωcom 7848   ↑_m cmap 8810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5103  df-opab 5165  df-cnv 5657  df-dm 5659  df-rn 5660  df-suc 6354  df-iota 6479  df-fv 6531  df-ov 7401

This theorem is referenced by:  ssfin3ds  10289  fin23lem17  10297  fin23lem39  10309  fin23lem40  10310  isf32lem12  10323  isfin3-3  10327