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Theorem isfin3ds 9343
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 5951 . . . . . . . . 9 (𝑏 = 𝑥 → suc 𝑏 = suc 𝑥)
21fveq2d 6356 . . . . . . . 8 (𝑏 = 𝑥 → (𝑎‘suc 𝑏) = (𝑎‘suc 𝑥))
3 fveq2 6352 . . . . . . . 8 (𝑏 = 𝑥 → (𝑎𝑏) = (𝑎𝑥))
42, 3sseq12d 3775 . . . . . . 7 (𝑏 = 𝑥 → ((𝑎‘suc 𝑏) ⊆ (𝑎𝑏) ↔ (𝑎‘suc 𝑥) ⊆ (𝑎𝑥)))
54cbvralv 3310 . . . . . 6 (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) ↔ ∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥))
6 fveq1 6351 . . . . . . . 8 (𝑎 = 𝑓 → (𝑎‘suc 𝑥) = (𝑓‘suc 𝑥))
7 fveq1 6351 . . . . . . . 8 (𝑎 = 𝑓 → (𝑎𝑥) = (𝑓𝑥))
86, 7sseq12d 3775 . . . . . . 7 (𝑎 = 𝑓 → ((𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ (𝑓‘suc 𝑥) ⊆ (𝑓𝑥)))
98ralbidv 3124 . . . . . 6 (𝑎 = 𝑓 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥)))
105, 9syl5bb 272 . . . . 5 (𝑎 = 𝑓 → (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥)))
11 rneq 5506 . . . . . . 7 (𝑎 = 𝑓 → ran 𝑎 = ran 𝑓)
1211inteqd 4632 . . . . . 6 (𝑎 = 𝑓 ran 𝑎 = ran 𝑓)
1312, 11eleq12d 2833 . . . . 5 (𝑎 = 𝑓 → ( ran 𝑎 ∈ ran 𝑎 ran 𝑓 ∈ ran 𝑓))
1410, 13imbi12d 333 . . . 4 (𝑎 = 𝑓 → ((∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) → ran 𝑎 ∈ ran 𝑎) ↔ (∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
1514cbvralv 3310 . . 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 4305 . . . . 5 (𝑔 = 𝐴 → 𝒫 𝑔 = 𝒫 𝐴)
1716oveq1d 6828 . . . 4 (𝑔 = 𝐴 → (𝒫 𝑔𝑚 ω) = (𝒫 𝐴𝑚 ω))
1817raleqdv 3283 . . 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 272 . 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 3493 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 196   = wceq 1632  wcel 2139  {cab 2746  wral 3050  wss 3715  𝒫 cpw 4302   cint 4627  ran crn 5267  suc csuc 5886  cfv 6049  (class class class)co 6813  ωcom 7230  𝑚 cmap 8023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-cnv 5274  df-dm 5276  df-rn 5277  df-suc 5890  df-iota 6012  df-fv 6057  df-ov 6816
This theorem is referenced by:  ssfin3ds  9344  fin23lem17  9352  fin23lem39  9364  fin23lem40  9365  isf32lem12  9378  isfin3-3  9382
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