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Theorem fin23lem40 9767
Description: Lemma for fin23 9805. 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 8420 . . . 4 (𝑓 ∈ (𝒫 𝐴m ω) → 𝑓:ω⟶𝒫 𝐴)
2 simpl 486 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → 𝐴 ∈ FinII)
3 frn 6509 . . . . . . 7 (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ⊆ 𝒫 𝐴)
43ad2antrl 727 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → ran 𝑓 ⊆ 𝒫 𝐴)
5 fdm 6511 . . . . . . . . 9 (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 = ω)
6 peano1 7592 . . . . . . . . . 10 ∅ ∈ ω
7 ne0i 4283 . . . . . . . . . 10 (∅ ∈ ω → ω ≠ ∅)
86, 7mp1i 13 . . . . . . . . 9 (𝑓:ω⟶𝒫 𝐴 → ω ≠ ∅)
95, 8eqnetrd 3081 . . . . . . . 8 (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 ≠ ∅)
10 dm0rn0 5783 . . . . . . . . 9 (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
1110necon3bii 3066 . . . . . . . 8 (dom 𝑓 ≠ ∅ ↔ ran 𝑓 ≠ ∅)
129, 11sylib 221 . . . . . . 7 (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ≠ ∅)
1312ad2antrl 727 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → ran 𝑓 ≠ ∅)
14 ffn 6503 . . . . . . . . 9 (𝑓:ω⟶𝒫 𝐴𝑓 Fn ω)
1514ad2antrl 727 . . . . . . . 8 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → 𝑓 Fn ω)
16 sspss 4062 . . . . . . . . . . 11 ((𝑓‘suc 𝑏) ⊆ (𝑓𝑏) ↔ ((𝑓‘suc 𝑏) ⊊ (𝑓𝑏) ∨ (𝑓‘suc 𝑏) = (𝑓𝑏)))
17 fvex 6672 . . . . . . . . . . . . . 14 (𝑓𝑏) ∈ V
18 fvex 6672 . . . . . . . . . . . . . 14 (𝑓‘suc 𝑏) ∈ V
1917, 18brcnv 5741 . . . . . . . . . . . . 13 ((𝑓𝑏) [] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) [] (𝑓𝑏))
2017brrpss 7443 . . . . . . . . . . . . 13 ((𝑓‘suc 𝑏) [] (𝑓𝑏) ↔ (𝑓‘suc 𝑏) ⊊ (𝑓𝑏))
2119, 20bitri 278 . . . . . . . . . . . 12 ((𝑓𝑏) [] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) ⊊ (𝑓𝑏))
22 eqcom 2831 . . . . . . . . . . . 12 ((𝑓𝑏) = (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) = (𝑓𝑏))
2321, 22orbi12i 912 . . . . . . . . . . 11 (((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)) ↔ ((𝑓‘suc 𝑏) ⊊ (𝑓𝑏) ∨ (𝑓‘suc 𝑏) = (𝑓𝑏)))
2416, 23sylbb2 241 . . . . . . . . . 10 ((𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)))
2524ralimi 3155 . . . . . . . . 9 (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ∀𝑏 ∈ ω ((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)))
2625ad2antll 728 . . . . . . . 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 7444 . . . . . . . . . 10 [] Po ran 𝑓
28 cnvpo 6126 . . . . . . . . . 10 ( [] Po ran 𝑓 [] Po ran 𝑓)
2927, 28mpbi 233 . . . . . . . . 9 [] Po ran 𝑓
3029a1i 11 . . . . . . . 8 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → [] Po ran 𝑓)
31 sornom 9693 . . . . . . . 8 ((𝑓 Fn ω ∧ ∀𝑏 ∈ ω ((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)) ∧ [] Po ran 𝑓) → [] Or ran 𝑓)
3215, 26, 30, 31syl3anc 1368 . . . . . . 7 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → [] Or ran 𝑓)
33 cnvso 6127 . . . . . . 7 ( [] Or ran 𝑓 [] Or ran 𝑓)
3432, 33sylibr 237 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → [] Or ran 𝑓)
35 fin2i2 9734 . . . . . 6 (((𝐴 ∈ FinII ∧ ran 𝑓 ⊆ 𝒫 𝐴) ∧ (ran 𝑓 ≠ ∅ ∧ [] Or ran 𝑓)) → ran 𝑓 ∈ ran 𝑓)
362, 4, 13, 34, 35syl22anc 837 . . . . 5 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → ran 𝑓 ∈ ran 𝑓)
3736expr 460 . . . 4 ((𝐴 ∈ FinII𝑓:ω⟶𝒫 𝐴) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓))
381, 37sylan2 595 . . 3 ((𝐴 ∈ FinII𝑓 ∈ (𝒫 𝐴m ω)) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓))
3938ralrimiva 3177 . 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 9745 . 2 (𝐴 ∈ FinII → (𝐴𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓)))
4239, 41mpbird 260 1 (𝐴 ∈ FinII𝐴𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844   = wceq 1538  wcel 2115  {cab 2802  wne 3014  wral 3133  wss 3919  wpss 3920  c0 4276  𝒫 cpw 4522   cint 4863   class class class wbr 5053   Po wpo 5460   Or wor 5461  ccnv 5542  dom cdm 5543  ran crn 5544  suc csuc 6181   Fn wfn 6339  wf 6340  cfv 6344  (class class class)co 7146   [] crpss 7439  ωcom 7571  m cmap 8398  FinIIcfin2 9695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-int 4864  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-rpss 7440  df-om 7572  df-1st 7681  df-2nd 7682  df-map 8400  df-fin2 9702
This theorem is referenced by:  fin23  9805
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