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Theorem isfin3ds 9948
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 𝑔m ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) → ran 𝑎 ∈ ran 𝑎)}
Assertion
Ref Expression
isfin3ds (𝐴𝑉 → (𝐴𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → 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
StepHypRef Expression
1 suceq 6283 . . . . . . . . 9 (𝑏 = 𝑥 → suc 𝑏 = suc 𝑥)
21fveq2d 6726 . . . . . . . 8 (𝑏 = 𝑥 → (𝑎‘suc 𝑏) = (𝑎‘suc 𝑥))
3 fveq2 6722 . . . . . . . 8 (𝑏 = 𝑥 → (𝑎𝑏) = (𝑎𝑥))
42, 3sseq12d 3939 . . . . . . 7 (𝑏 = 𝑥 → ((𝑎‘suc 𝑏) ⊆ (𝑎𝑏) ↔ (𝑎‘suc 𝑥) ⊆ (𝑎𝑥)))
54cbvralvw 3363 . . . . . 6 (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) ↔ ∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥))
6 fveq1 6721 . . . . . . . 8 (𝑎 = 𝑓 → (𝑎‘suc 𝑥) = (𝑓‘suc 𝑥))
7 fveq1 6721 . . . . . . . 8 (𝑎 = 𝑓 → (𝑎𝑥) = (𝑓𝑥))
86, 7sseq12d 3939 . . . . . . 7 (𝑎 = 𝑓 → ((𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ (𝑓‘suc 𝑥) ⊆ (𝑓𝑥)))
98ralbidv 3118 . . . . . 6 (𝑎 = 𝑓 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥)))
105, 9syl5bb 286 . . . . 5 (𝑎 = 𝑓 → (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥)))
11 rneq 5810 . . . . . . 7 (𝑎 = 𝑓 → ran 𝑎 = ran 𝑓)
1211inteqd 4869 . . . . . 6 (𝑎 = 𝑓 ran 𝑎 = ran 𝑓)
1312, 11eleq12d 2832 . . . . 5 (𝑎 = 𝑓 → ( ran 𝑎 ∈ ran 𝑎 ran 𝑓 ∈ ran 𝑓))
1410, 13imbi12d 348 . . . 4 (𝑎 = 𝑓 → ((∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) → ran 𝑎 ∈ ran 𝑎) ↔ (∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
1514cbvralvw 3363 . . 3 (∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) → ran 𝑎 ∈ ran 𝑎) ↔ ∀𝑓 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓))
16 pweq 4534 . . . . 5 (𝑔 = 𝐴 → 𝒫 𝑔 = 𝒫 𝐴)
1716oveq1d 7233 . . . 4 (𝑔 = 𝐴 → (𝒫 𝑔m ω) = (𝒫 𝐴m ω))
1817raleqdv 3330 . . 3 (𝑔 = 𝐴 → (∀𝑓 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓) ↔ ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
1915, 18syl5bb 286 . 2 (𝑔 = 𝐴 → (∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) → ran 𝑎 ∈ ran 𝑎) ↔ ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
20 isfin3ds.f . 2 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) → ran 𝑎 ∈ ran 𝑎)}
2119, 20elab2g 3594 1 (𝐴𝑉 → (𝐴𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2110  {cab 2714  wral 3061  wss 3871  𝒫 cpw 4518   cint 4864  ran crn 5557  suc csuc 6220  cfv 6385  (class class class)co 7218  ωcom 7649  m cmap 8513
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 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rab 3070  df-v 3415  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-pw 4520  df-sn 4547  df-pr 4549  df-op 4553  df-uni 4825  df-int 4865  df-br 5059  df-opab 5121  df-cnv 5564  df-dm 5566  df-rn 5567  df-suc 6224  df-iota 6343  df-fv 6393  df-ov 7221
This theorem is referenced by:  ssfin3ds  9949  fin23lem17  9957  fin23lem39  9969  fin23lem40  9970  isf32lem12  9983  isfin3-3  9987
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