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Theorem isfin3ds 9011
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 5693 . . . . . . . . 9 (𝑏 = 𝑥 → suc 𝑏 = suc 𝑥)
21fveq2d 6092 . . . . . . . 8 (𝑏 = 𝑥 → (𝑎‘suc 𝑏) = (𝑎‘suc 𝑥))
3 fveq2 6088 . . . . . . . 8 (𝑏 = 𝑥 → (𝑎𝑏) = (𝑎𝑥))
42, 3sseq12d 3596 . . . . . . 7 (𝑏 = 𝑥 → ((𝑎‘suc 𝑏) ⊆ (𝑎𝑏) ↔ (𝑎‘suc 𝑥) ⊆ (𝑎𝑥)))
54cbvralv 3146 . . . . . 6 (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) ↔ ∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥))
6 fveq1 6087 . . . . . . . 8 (𝑎 = 𝑓 → (𝑎‘suc 𝑥) = (𝑓‘suc 𝑥))
7 fveq1 6087 . . . . . . . 8 (𝑎 = 𝑓 → (𝑎𝑥) = (𝑓𝑥))
86, 7sseq12d 3596 . . . . . . 7 (𝑎 = 𝑓 → ((𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ (𝑓‘suc 𝑥) ⊆ (𝑓𝑥)))
98ralbidv 2968 . . . . . 6 (𝑎 = 𝑓 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥)))
105, 9syl5bb 270 . . . . 5 (𝑎 = 𝑓 → (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥)))
11 rneq 5259 . . . . . . 7 (𝑎 = 𝑓 → ran 𝑎 = ran 𝑓)
1211inteqd 4409 . . . . . 6 (𝑎 = 𝑓 ran 𝑎 = ran 𝑓)
1312, 11eleq12d 2681 . . . . 5 (𝑎 = 𝑓 → ( ran 𝑎 ∈ ran 𝑎 ran 𝑓 ∈ ran 𝑓))
1410, 13imbi12d 332 . . . 4 (𝑎 = 𝑓 → ((∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) → ran 𝑎 ∈ ran 𝑎) ↔ (∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
1514cbvralv 3146 . . 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 4110 . . . . 5 (𝑔 = 𝐴 → 𝒫 𝑔 = 𝒫 𝐴)
1716oveq1d 6542 . . . 4 (𝑔 = 𝐴 → (𝒫 𝑔𝑚 ω) = (𝒫 𝐴𝑚 ω))
1817raleqdv 3120 . . 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 270 . 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 3321 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 194   = wceq 1474  wcel 1976  {cab 2595  wral 2895  wss 3539  𝒫 cpw 4107   cint 4404  ran crn 5029  suc csuc 5628  cfv 5790  (class class class)co 6527  ωcom 6934  𝑚 cmap 7721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-cnv 5036  df-dm 5038  df-rn 5039  df-suc 5632  df-iota 5754  df-fv 5798  df-ov 6530
This theorem is referenced by:  ssfin3ds  9012  fin23lem17  9020  fin23lem39  9032  fin23lem40  9033  isf32lem12  9046  isfin3-3  9050
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