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Theorem isfin3ds 10324
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 6431 . . . . . . . . 9 (𝑏 = 𝑥 → suc 𝑏 = suc 𝑥)
21fveq2d 6896 . . . . . . . 8 (𝑏 = 𝑥 → (𝑎‘suc 𝑏) = (𝑎‘suc 𝑥))
3 fveq2 6892 . . . . . . . 8 (𝑏 = 𝑥 → (𝑎𝑏) = (𝑎𝑥))
42, 3sseq12d 4016 . . . . . . 7 (𝑏 = 𝑥 → ((𝑎‘suc 𝑏) ⊆ (𝑎𝑏) ↔ (𝑎‘suc 𝑥) ⊆ (𝑎𝑥)))
54cbvralvw 3235 . . . . . 6 (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) ↔ ∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥))
6 fveq1 6891 . . . . . . . 8 (𝑎 = 𝑓 → (𝑎‘suc 𝑥) = (𝑓‘suc 𝑥))
7 fveq1 6891 . . . . . . . 8 (𝑎 = 𝑓 → (𝑎𝑥) = (𝑓𝑥))
86, 7sseq12d 4016 . . . . . . 7 (𝑎 = 𝑓 → ((𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ (𝑓‘suc 𝑥) ⊆ (𝑓𝑥)))
98ralbidv 3178 . . . . . 6 (𝑎 = 𝑓 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥)))
105, 9bitrid 283 . . . . 5 (𝑎 = 𝑓 → (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥)))
11 rneq 5936 . . . . . . 7 (𝑎 = 𝑓 → ran 𝑎 = ran 𝑓)
1211inteqd 4956 . . . . . 6 (𝑎 = 𝑓 ran 𝑎 = ran 𝑓)
1312, 11eleq12d 2828 . . . . 5 (𝑎 = 𝑓 → ( ran 𝑎 ∈ ran 𝑎 ran 𝑓 ∈ ran 𝑓))
1410, 13imbi12d 345 . . . 4 (𝑎 = 𝑓 → ((∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) → ran 𝑎 ∈ ran 𝑎) ↔ (∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
1514cbvralvw 3235 . . 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 4617 . . . . 5 (𝑔 = 𝐴 → 𝒫 𝑔 = 𝒫 𝐴)
1716oveq1d 7424 . . . 4 (𝑔 = 𝐴 → (𝒫 𝑔m ω) = (𝒫 𝐴m ω))
1817raleqdv 3326 . . 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, 18bitrid 283 . 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 3671 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 205   = wceq 1542  wcel 2107  {cab 2710  wral 3062  wss 3949  𝒫 cpw 4603   cint 4951  ran crn 5678  suc csuc 6367  cfv 6544  (class class class)co 7409  ωcom 7855  m cmap 8820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-cnv 5685  df-dm 5687  df-rn 5688  df-suc 6371  df-iota 6496  df-fv 6552  df-ov 7412
This theorem is referenced by:  ssfin3ds  10325  fin23lem17  10333  fin23lem39  10345  fin23lem40  10346  isf32lem12  10359  isfin3-3  10363
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