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Theorem fin23lem40 9211
Description: Lemma for fin23 9249. FinII sets satisfy the descending chain condition. (Contributed by Stefan O'Rear, 3-Nov-2014.)
Hypothesis
Ref Expression
fin23lem40.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
Assertion
Ref Expression
fin23lem40 (𝐴 ∈ FinII𝐴𝐹)
Distinct variable groups:   𝑔,𝑎,𝑥,𝐴   𝐹,𝑎
Allowed substitution hints:   𝐹(𝑥,𝑔)

Proof of Theorem fin23lem40
Dummy variables 𝑏 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elmapi 7921 . . . 4 (𝑓 ∈ (𝒫 𝐴𝑚 ω) → 𝑓:ω⟶𝒫 𝐴)
2 simpl 472 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → 𝐴 ∈ FinII)
3 frn 6091 . . . . . . 7 (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ⊆ 𝒫 𝐴)
43ad2antrl 764 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → ran 𝑓 ⊆ 𝒫 𝐴)
5 fdm 6089 . . . . . . . . 9 (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 = ω)
6 peano1 7127 . . . . . . . . . 10 ∅ ∈ ω
7 ne0i 3954 . . . . . . . . . 10 (∅ ∈ ω → ω ≠ ∅)
86, 7mp1i 13 . . . . . . . . 9 (𝑓:ω⟶𝒫 𝐴 → ω ≠ ∅)
95, 8eqnetrd 2890 . . . . . . . 8 (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 ≠ ∅)
10 dm0rn0 5374 . . . . . . . . 9 (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
1110necon3bii 2875 . . . . . . . 8 (dom 𝑓 ≠ ∅ ↔ ran 𝑓 ≠ ∅)
129, 11sylib 208 . . . . . . 7 (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ≠ ∅)
1312ad2antrl 764 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → ran 𝑓 ≠ ∅)
14 ffn 6083 . . . . . . . . 9 (𝑓:ω⟶𝒫 𝐴𝑓 Fn ω)
1514ad2antrl 764 . . . . . . . 8 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → 𝑓 Fn ω)
16 sspss 3739 . . . . . . . . . . 11 ((𝑓‘suc 𝑏) ⊆ (𝑓𝑏) ↔ ((𝑓‘suc 𝑏) ⊊ (𝑓𝑏) ∨ (𝑓‘suc 𝑏) = (𝑓𝑏)))
17 fvex 6239 . . . . . . . . . . . . . 14 (𝑓𝑏) ∈ V
18 fvex 6239 . . . . . . . . . . . . . 14 (𝑓‘suc 𝑏) ∈ V
1917, 18brcnv 5337 . . . . . . . . . . . . 13 ((𝑓𝑏) [] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) [] (𝑓𝑏))
2017brrpss 6982 . . . . . . . . . . . . 13 ((𝑓‘suc 𝑏) [] (𝑓𝑏) ↔ (𝑓‘suc 𝑏) ⊊ (𝑓𝑏))
2119, 20bitri 264 . . . . . . . . . . . 12 ((𝑓𝑏) [] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) ⊊ (𝑓𝑏))
22 eqcom 2658 . . . . . . . . . . . 12 ((𝑓𝑏) = (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) = (𝑓𝑏))
2321, 22orbi12i 542 . . . . . . . . . . 11 (((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)) ↔ ((𝑓‘suc 𝑏) ⊊ (𝑓𝑏) ∨ (𝑓‘suc 𝑏) = (𝑓𝑏)))
2416, 23sylbb2 228 . . . . . . . . . 10 ((𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)))
2524ralimi 2981 . . . . . . . . 9 (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ∀𝑏 ∈ ω ((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)))
2625ad2antll 765 . . . . . . . 8 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → ∀𝑏 ∈ ω ((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)))
27 porpss 6983 . . . . . . . . . 10 [] Po ran 𝑓
28 cnvpo 5711 . . . . . . . . . 10 ( [] Po ran 𝑓 [] Po ran 𝑓)
2927, 28mpbi 220 . . . . . . . . 9 [] Po ran 𝑓
3029a1i 11 . . . . . . . 8 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → [] Po ran 𝑓)
31 sornom 9137 . . . . . . . 8 ((𝑓 Fn ω ∧ ∀𝑏 ∈ ω ((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)) ∧ [] Po ran 𝑓) → [] Or ran 𝑓)
3215, 26, 30, 31syl3anc 1366 . . . . . . 7 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → [] Or ran 𝑓)
33 cnvso 5712 . . . . . . 7 ( [] Or ran 𝑓 [] Or ran 𝑓)
3432, 33sylibr 224 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → [] Or ran 𝑓)
35 fin2i2 9178 . . . . . 6 (((𝐴 ∈ FinII ∧ ran 𝑓 ⊆ 𝒫 𝐴) ∧ (ran 𝑓 ≠ ∅ ∧ [] Or ran 𝑓)) → ran 𝑓 ∈ ran 𝑓)
362, 4, 13, 34, 35syl22anc 1367 . . . . 5 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → ran 𝑓 ∈ ran 𝑓)
3736expr 642 . . . 4 ((𝐴 ∈ FinII𝑓:ω⟶𝒫 𝐴) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓))
381, 37sylan2 490 . . 3 ((𝐴 ∈ FinII𝑓 ∈ (𝒫 𝐴𝑚 ω)) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓))
3938ralrimiva 2995 . 2 (𝐴 ∈ FinII → ∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓))
40 fin23lem40.f . . 3 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
4140isfin3ds 9189 . 2 (𝐴 ∈ FinII → (𝐴𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴𝑚 ω)(∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓)))
4239, 41mpbird 247 1 (𝐴 ∈ FinII𝐴𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  wss 3607  wpss 3608  c0 3948  𝒫 cpw 4191   cint 4507   class class class wbr 4685   Po wpo 5062   Or wor 5063  ccnv 5142  dom cdm 5143  ran crn 5144  suc csuc 5763   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690   [] crpss 6978  ωcom 7107  𝑚 cmap 7899  FinIIcfin2 9139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-rpss 6979  df-om 7108  df-1st 7210  df-2nd 7211  df-map 7901  df-fin2 9146
This theorem is referenced by:  fin23  9249
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