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Theorem fin23lem40 10388
Description: Lemma for fin23 10426. 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 8887 . . . 4 (𝑓 ∈ (𝒫 𝐴m ω) → 𝑓:ω⟶𝒫 𝐴)
2 simpl 482 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → 𝐴 ∈ FinII)
3 frn 6743 . . . . . . 7 (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ⊆ 𝒫 𝐴)
43ad2antrl 728 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → ran 𝑓 ⊆ 𝒫 𝐴)
5 fdm 6745 . . . . . . . . 9 (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 = ω)
6 peano1 7910 . . . . . . . . . 10 ∅ ∈ ω
7 ne0i 4346 . . . . . . . . . 10 (∅ ∈ ω → ω ≠ ∅)
86, 7mp1i 13 . . . . . . . . 9 (𝑓:ω⟶𝒫 𝐴 → ω ≠ ∅)
95, 8eqnetrd 3005 . . . . . . . 8 (𝑓:ω⟶𝒫 𝐴 → dom 𝑓 ≠ ∅)
10 dm0rn0 5937 . . . . . . . . 9 (dom 𝑓 = ∅ ↔ ran 𝑓 = ∅)
1110necon3bii 2990 . . . . . . . 8 (dom 𝑓 ≠ ∅ ↔ ran 𝑓 ≠ ∅)
129, 11sylib 218 . . . . . . 7 (𝑓:ω⟶𝒫 𝐴 → ran 𝑓 ≠ ∅)
1312ad2antrl 728 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → ran 𝑓 ≠ ∅)
14 ffn 6736 . . . . . . . . 9 (𝑓:ω⟶𝒫 𝐴𝑓 Fn ω)
1514ad2antrl 728 . . . . . . . 8 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → 𝑓 Fn ω)
16 sspss 4111 . . . . . . . . . . 11 ((𝑓‘suc 𝑏) ⊆ (𝑓𝑏) ↔ ((𝑓‘suc 𝑏) ⊊ (𝑓𝑏) ∨ (𝑓‘suc 𝑏) = (𝑓𝑏)))
17 fvex 6919 . . . . . . . . . . . . . 14 (𝑓𝑏) ∈ V
18 fvex 6919 . . . . . . . . . . . . . 14 (𝑓‘suc 𝑏) ∈ V
1917, 18brcnv 5895 . . . . . . . . . . . . 13 ((𝑓𝑏) [] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) [] (𝑓𝑏))
2017brrpss 7744 . . . . . . . . . . . . 13 ((𝑓‘suc 𝑏) [] (𝑓𝑏) ↔ (𝑓‘suc 𝑏) ⊊ (𝑓𝑏))
2119, 20bitri 275 . . . . . . . . . . . 12 ((𝑓𝑏) [] (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) ⊊ (𝑓𝑏))
22 eqcom 2741 . . . . . . . . . . . 12 ((𝑓𝑏) = (𝑓‘suc 𝑏) ↔ (𝑓‘suc 𝑏) = (𝑓𝑏))
2321, 22orbi12i 914 . . . . . . . . . . 11 (((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)) ↔ ((𝑓‘suc 𝑏) ⊊ (𝑓𝑏) ∨ (𝑓‘suc 𝑏) = (𝑓𝑏)))
2416, 23sylbb2 238 . . . . . . . . . 10 ((𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)))
2524ralimi 3080 . . . . . . . . 9 (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ∀𝑏 ∈ ω ((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)))
2625ad2antll 729 . . . . . . . 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 7745 . . . . . . . . . 10 [] Po ran 𝑓
28 cnvpo 6308 . . . . . . . . . 10 ( [] Po ran 𝑓 [] Po ran 𝑓)
2927, 28mpbi 230 . . . . . . . . 9 [] Po ran 𝑓
3029a1i 11 . . . . . . . 8 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → [] Po ran 𝑓)
31 sornom 10314 . . . . . . . 8 ((𝑓 Fn ω ∧ ∀𝑏 ∈ ω ((𝑓𝑏) [] (𝑓‘suc 𝑏) ∨ (𝑓𝑏) = (𝑓‘suc 𝑏)) ∧ [] Po ran 𝑓) → [] Or ran 𝑓)
3215, 26, 30, 31syl3anc 1370 . . . . . . 7 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → [] Or ran 𝑓)
33 cnvso 6309 . . . . . . 7 ( [] Or ran 𝑓 [] Or ran 𝑓)
3432, 33sylibr 234 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → [] Or ran 𝑓)
35 fin2i2 10355 . . . . . 6 (((𝐴 ∈ FinII ∧ ran 𝑓 ⊆ 𝒫 𝐴) ∧ (ran 𝑓 ≠ ∅ ∧ [] Or ran 𝑓)) → ran 𝑓 ∈ ran 𝑓)
362, 4, 13, 34, 35syl22anc 839 . . . . 5 ((𝐴 ∈ FinII ∧ (𝑓:ω⟶𝒫 𝐴 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏))) → ran 𝑓 ∈ ran 𝑓)
3736expr 456 . . . 4 ((𝐴 ∈ FinII𝑓:ω⟶𝒫 𝐴) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓))
381, 37sylan2 593 . . 3 ((𝐴 ∈ FinII𝑓 ∈ (𝒫 𝐴m ω)) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓))
3938ralrimiva 3143 . 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 10366 . 2 (𝐴 ∈ FinII → (𝐴𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐴m ω)(∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓𝑏) → ran 𝑓 ∈ ran 𝑓)))
4239, 41mpbird 257 1 (𝐴 ∈ FinII𝐴𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1536  wcel 2105  {cab 2711  wne 2937  wral 3058  wss 3962  wpss 3963  c0 4338  𝒫 cpw 4604   cint 4950   class class class wbr 5147   Po wpo 5594   Or wor 5595  ccnv 5687  dom cdm 5688  ran crn 5689  suc csuc 6387   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430   [] crpss 7740  ωcom 7886  m cmap 8864  FinIIcfin2 10316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-rpss 7741  df-om 7887  df-1st 8012  df-2nd 8013  df-map 8866  df-fin2 10323
This theorem is referenced by:  fin23  10426
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