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Theorem isfin3ds 9404
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 5973 . . . . . . . . 9 (𝑏 = 𝑥 → suc 𝑏 = suc 𝑥)
21fveq2d 6379 . . . . . . . 8 (𝑏 = 𝑥 → (𝑎‘suc 𝑏) = (𝑎‘suc 𝑥))
3 fveq2 6375 . . . . . . . 8 (𝑏 = 𝑥 → (𝑎𝑏) = (𝑎𝑥))
42, 3sseq12d 3794 . . . . . . 7 (𝑏 = 𝑥 → ((𝑎‘suc 𝑏) ⊆ (𝑎𝑏) ↔ (𝑎‘suc 𝑥) ⊆ (𝑎𝑥)))
54cbvralv 3319 . . . . . 6 (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) ↔ ∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥))
6 fveq1 6374 . . . . . . . 8 (𝑎 = 𝑓 → (𝑎‘suc 𝑥) = (𝑓‘suc 𝑥))
7 fveq1 6374 . . . . . . . 8 (𝑎 = 𝑓 → (𝑎𝑥) = (𝑓𝑥))
86, 7sseq12d 3794 . . . . . . 7 (𝑎 = 𝑓 → ((𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ (𝑓‘suc 𝑥) ⊆ (𝑓𝑥)))
98ralbidv 3133 . . . . . 6 (𝑎 = 𝑓 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥)))
105, 9syl5bb 274 . . . . 5 (𝑎 = 𝑓 → (∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) ↔ ∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥)))
11 rneq 5519 . . . . . . 7 (𝑎 = 𝑓 → ran 𝑎 = ran 𝑓)
1211inteqd 4638 . . . . . 6 (𝑎 = 𝑓 ran 𝑎 = ran 𝑓)
1312, 11eleq12d 2838 . . . . 5 (𝑎 = 𝑓 → ( ran 𝑎 ∈ ran 𝑎 ran 𝑓 ∈ ran 𝑓))
1410, 13imbi12d 335 . . . 4 (𝑎 = 𝑓 → ((∀𝑏 ∈ ω (𝑎‘suc 𝑏) ⊆ (𝑎𝑏) → ran 𝑎 ∈ ran 𝑎) ↔ (∀𝑥 ∈ ω (𝑓‘suc 𝑥) ⊆ (𝑓𝑥) → ran 𝑓 ∈ ran 𝑓)))
1514cbvralv 3319 . . 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 4318 . . . . 5 (𝑔 = 𝐴 → 𝒫 𝑔 = 𝒫 𝐴)
1716oveq1d 6857 . . . 4 (𝑔 = 𝐴 → (𝒫 𝑔𝑚 ω) = (𝒫 𝐴𝑚 ω))
1817raleqdv 3292 . . 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 274 . 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 3508 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 197   = wceq 1652  wcel 2155  {cab 2751  wral 3055  wss 3732  𝒫 cpw 4315   cint 4633  ran crn 5278  suc csuc 5910  cfv 6068  (class class class)co 6842  ωcom 7263  𝑚 cmap 8060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-int 4634  df-br 4810  df-opab 4872  df-cnv 5285  df-dm 5287  df-rn 5288  df-suc 5914  df-iota 6031  df-fv 6076  df-ov 6845
This theorem is referenced by:  ssfin3ds  9405  fin23lem17  9413  fin23lem39  9425  fin23lem40  9426  isf32lem12  9439  isfin3-3  9443
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