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Theorem fin23lem40 9761
Description: Lemma for fin23 9799. 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 8417 . . . 4 (𝑓 ∈ (𝒫 𝐴m ω) → 𝑓:ω⟶𝒫 𝐴)
2 simpl 483 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → 𝐴 ∈ FinII)
3 frn 6513 . . . . . . 7 (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ⊆ 𝒫 𝐴)
43ad2antrl 724 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → ran 𝑓 ⊆ 𝒫 𝐴)
5 fdm 6515 . . . . . . . . 9 (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 = ω)
6 peano1 7590 . . . . . . . . . 10 ∅ ∈ ω
7 ne0i 4297 . . . . . . . . . 10 (∅ ∈ ω → ω ≠ ∅)
86, 7mp1i 13 . . . . . . . . 9 (𝑓:ω⟶𝒫 𝐴 → ω ≠ ∅)
95, 8eqnetrd 3080 . . . . . . . 8 (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 ≠ ∅)
10 dm0rn0 5788 . . . . . . . . 9 (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
1110necon3bii 3065 . . . . . . . 8 (dom 𝑓 ≠ ∅ ↔ ran 𝑓 ≠ ∅)
129, 11sylib 219 . . . . . . 7 (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ≠ ∅)
1312ad2antrl 724 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → ran 𝑓 ≠ ∅)
14 ffn 6507 . . . . . . . . 9 (𝑓:ω⟶𝒫 𝐴𝑓 Fn ω)
1514ad2antrl 724 . . . . . . . 8 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → 𝑓 Fn ω)
16 sspss 4073 . . . . . . . . . . 11 ((𝑓‘suc 𝑏) ⊆ (𝑓𝑏) ↔ ((𝑓‘suc 𝑏) ⊊ (𝑓𝑏) ∨ (𝑓‘suc 𝑏) = (𝑓𝑏)))
17 fvex 6676 . . . . . . . . . . . . . 14 (𝑓𝑏) ∈ V
18 fvex 6676 . . . . . . . . . . . . . 14 (𝑓‘suc 𝑏) ∈ V
1917, 18brcnv 5746 . . . . . . . . . . . . 13 ((𝑓𝑏) [] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) [] (𝑓𝑏))
2017brrpss 7441 . . . . . . . . . . . . 13 ((𝑓‘suc 𝑏) [] (𝑓𝑏) ↔ (𝑓‘suc 𝑏) ⊊ (𝑓𝑏))
2119, 20bitri 276 . . . . . . . . . . . 12 ((𝑓𝑏) [] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) ⊊ (𝑓𝑏))
22 eqcom 2825 . . . . . . . . . . . 12 ((𝑓𝑏) = (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) = (𝑓𝑏))
2321, 22orbi12i 908 . . . . . . . . . . 11 (((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)) ↔ ((𝑓‘suc 𝑏) ⊊ (𝑓𝑏) ∨ (𝑓‘suc 𝑏) = (𝑓𝑏)))
2416, 23sylbb2 239 . . . . . . . . . 10 ((𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)))
2524ralimi 3157 . . . . . . . . 9 (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ∀𝑏 ∈ ω ((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)))
2625ad2antll 725 . . . . . . . 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 7442 . . . . . . . . . 10 [] Po ran 𝑓
28 cnvpo 6131 . . . . . . . . . 10 ( [] Po ran 𝑓 [] Po ran 𝑓)
2927, 28mpbi 231 . . . . . . . . 9 [] Po ran 𝑓
3029a1i 11 . . . . . . . 8 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → [] Po ran 𝑓)
31 sornom 9687 . . . . . . . 8 ((𝑓 Fn ω ∧ ∀𝑏 ∈ ω ((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)) ∧ [] Po ran 𝑓) → [] Or ran 𝑓)
3215, 26, 30, 31syl3anc 1363 . . . . . . 7 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → [] Or ran 𝑓)
33 cnvso 6132 . . . . . . 7 ( [] Or ran 𝑓 [] Or ran 𝑓)
3432, 33sylibr 235 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → [] Or ran 𝑓)
35 fin2i2 9728 . . . . . 6 (((𝐴 ∈ FinII ∧ ran 𝑓 ⊆ 𝒫 𝐴) ∧ (ran 𝑓 ≠ ∅ ∧ [] Or ran 𝑓)) → ran 𝑓 ∈ ran 𝑓)
362, 4, 13, 34, 35syl22anc 834 . . . . 5 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → ran 𝑓 ∈ ran 𝑓)
3736expr 457 . . . 4 ((𝐴 ∈ FinII𝑓:ω⟶𝒫 𝐴) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓))
381, 37sylan2 592 . . 3 ((𝐴 ∈ FinII𝑓 ∈ (𝒫 𝐴m ω)) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓))
3938ralrimiva 3179 . 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 9739 . 2 (𝐴 ∈ FinII → (𝐴𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓)))
4239, 41mpbird 258 1 (𝐴 ∈ FinII𝐴𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  {cab 2796  wne 3013  wral 3135  wss 3933  wpss 3934  c0 4288  𝒫 cpw 4535   cint 4867   class class class wbr 5057   Po wpo 5465   Or wor 5466  ccnv 5547  dom cdm 5548  ran crn 5549  suc csuc 6186   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145   [] crpss 7437  ωcom 7569  m cmap 8395  FinIIcfin2 9689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-rpss 7438  df-om 7570  df-1st 7678  df-2nd 7679  df-map 8397  df-fin2 9696
This theorem is referenced by:  fin23  9799
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