Theorem isfin3ds 10370

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 6449	. . . . . . . . 9 ⊢ (𝑏 = 𝑥 → suc 𝑏 = suc 𝑥)
2	1	fveq2d 6909	. . . . . . . 8 ⊢ (𝑏 = 𝑥 → (𝑎‘suc 𝑏) = (𝑎‘suc 𝑥))
3		fveq2 6905	. . . . . . . 8 ⊢ (𝑏 = 𝑥 → (𝑎‘𝑏) = (𝑎‘𝑥))
4	2, 3	sseq12d 4016	. . . . . . 7 ⊢ (𝑏 = 𝑥 → ((𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) ↔ (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥)))
5	4	cbvralvw 3236	. . . . . 6 ⊢ (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) ↔ ∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥))
6		fveq1 6904	. . . . . . . 8 ⊢ (𝑎 = 𝑓 → (𝑎‘suc 𝑥) = (𝑓‘suc 𝑥))
7		fveq1 6904	. . . . . . . 8 ⊢ (𝑎 = 𝑓 → (𝑎‘𝑥) = (𝑓‘𝑥))
8	6, 7	sseq12d 4016	. . . . . . 7 ⊢ (𝑎 = 𝑓 → ((𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥)))
9	8	ralbidv 3177	. . . . . 6 ⊢ (𝑎 = 𝑓 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥)))
10	5, 9	bitrid 283	. . . . 5 ⊢ (𝑎 = 𝑓 → (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎‘𝑏) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓‘𝑥)))
11		rneq 5946	. . . . . . 7 ⊢ (𝑎 = 𝑓 → ran 𝑎 = ran 𝑓)
12	11	inteqd 4950	. . . . . 6 ⊢ (𝑎 = 𝑓 → ∩ ran 𝑎 = ∩ ran 𝑓)
13	12, 11	eleq12d 2834	. . . . 5 ⊢ (𝑎 = 𝑓 → (∩ ran 𝑎 ∈ ran 𝑎 ↔ ∩ ran 𝑓 ∈ ran 𝑓))
14	10, 13	imbi12d 344	. . . 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 3236	. . 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 4613	. . . . 5 ⊢ (𝑔 = 𝐴 → 𝒫 𝑔 = 𝒫 𝐴)
17	16	oveq1d 7447	. . . 4 ⊢ (𝑔 = 𝐴 → (𝒫 𝑔 ↑m ω) = (𝒫 𝐴 ↑m ω))
18	17	raleqdv 3325	. . 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 283	. 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 3679	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 206   = wceq 1539   ∈ wcel 2107  {cab 2713  ∀wral 3060   ⊆ wss 3950  𝒫 cpw 4599  ∩ cint 4945  ran crn 5685  suc csuc 6385  ‘cfv 6560  (class class class)co 7432  ωcom 7888   ↑_m cmap 8867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-cnv 5692  df-dm 5694  df-rn 5695  df-suc 6389  df-iota 6513  df-fv 6568  df-ov 7435

This theorem is referenced by:  ssfin3ds  10371  fin23lem17  10379  fin23lem39  10391  fin23lem40  10392  isf32lem12  10405  isfin3-3  10409