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Theorem fin23lem40 10350
Description: Lemma for fin23 10388. FinII sets satisfy the descending chain condition. (Contributed by Stefan O'Rear, 3-Nov-2014.)
Hypothesis
Ref Expression
fin23lem40.f 𝐹 = {𝑔 ∣ βˆ€π‘Ž ∈ (𝒫 𝑔 ↑m Ο‰)(βˆ€π‘₯ ∈ Ο‰ (π‘Žβ€˜suc π‘₯) βŠ† (π‘Žβ€˜π‘₯) β†’ ∩ ran π‘Ž ∈ ran π‘Ž)}
Assertion
Ref Expression
fin23lem40 (𝐴 ∈ FinII β†’ 𝐴 ∈ 𝐹)
Distinct variable groups:   𝑔,π‘Ž,π‘₯,𝐴   𝐹,π‘Ž
Allowed substitution hints:   𝐹(π‘₯,𝑔)

Proof of Theorem fin23lem40
Dummy variables 𝑏 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elmapi 8847 . . . 4 (𝑓 ∈ (𝒫 𝐴 ↑m Ο‰) β†’ 𝑓:Ο‰βŸΆπ’« 𝐴)
2 simpl 482 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:Ο‰βŸΆπ’« 𝐴 ∧ βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘))) β†’ 𝐴 ∈ FinII)
3 frn 6724 . . . . . . 7 (𝑓:Ο‰βŸΆπ’« 𝐴 β†’ ran 𝑓 βŠ† 𝒫 𝐴)
43ad2antrl 725 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:Ο‰βŸΆπ’« 𝐴 ∧ βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘))) β†’ ran 𝑓 βŠ† 𝒫 𝐴)
5 fdm 6726 . . . . . . . . 9 (𝑓:Ο‰βŸΆπ’« 𝐴 β†’ dom 𝑓 = Ο‰)
6 peano1 7883 . . . . . . . . . 10 βˆ… ∈ Ο‰
7 ne0i 4334 . . . . . . . . . 10 (βˆ… ∈ Ο‰ β†’ Ο‰ β‰  βˆ…)
86, 7mp1i 13 . . . . . . . . 9 (𝑓:Ο‰βŸΆπ’« 𝐴 β†’ Ο‰ β‰  βˆ…)
95, 8eqnetrd 3007 . . . . . . . 8 (𝑓:Ο‰βŸΆπ’« 𝐴 β†’ dom 𝑓 β‰  βˆ…)
10 dm0rn0 5924 . . . . . . . . 9 (dom 𝑓 = βˆ… ↔ ran 𝑓 = βˆ…)
1110necon3bii 2992 . . . . . . . 8 (dom 𝑓 β‰  βˆ… ↔ ran 𝑓 β‰  βˆ…)
129, 11sylib 217 . . . . . . 7 (𝑓:Ο‰βŸΆπ’« 𝐴 β†’ ran 𝑓 β‰  βˆ…)
1312ad2antrl 725 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:Ο‰βŸΆπ’« 𝐴 ∧ βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘))) β†’ ran 𝑓 β‰  βˆ…)
14 ffn 6717 . . . . . . . . 9 (𝑓:Ο‰βŸΆπ’« 𝐴 β†’ 𝑓 Fn Ο‰)
1514ad2antrl 725 . . . . . . . 8 ((𝐴 ∈ FinII ∧ (𝑓:Ο‰βŸΆπ’« 𝐴 ∧ βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘))) β†’ 𝑓 Fn Ο‰)
16 sspss 4099 . . . . . . . . . . 11 ((π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘) ↔ ((π‘“β€˜suc 𝑏) ⊊ (π‘“β€˜π‘) ∨ (π‘“β€˜suc 𝑏) = (π‘“β€˜π‘)))
17 fvex 6904 . . . . . . . . . . . . . 14 (π‘“β€˜π‘) ∈ V
18 fvex 6904 . . . . . . . . . . . . . 14 (π‘“β€˜suc 𝑏) ∈ V
1917, 18brcnv 5882 . . . . . . . . . . . . 13 ((π‘“β€˜π‘)β—‘ [⊊] (π‘“β€˜suc 𝑏) ↔ (π‘“β€˜suc 𝑏) [⊊] (π‘“β€˜π‘))
2017brrpss 7720 . . . . . . . . . . . . 13 ((π‘“β€˜suc 𝑏) [⊊] (π‘“β€˜π‘) ↔ (π‘“β€˜suc 𝑏) ⊊ (π‘“β€˜π‘))
2119, 20bitri 275 . . . . . . . . . . . 12 ((π‘“β€˜π‘)β—‘ [⊊] (π‘“β€˜suc 𝑏) ↔ (π‘“β€˜suc 𝑏) ⊊ (π‘“β€˜π‘))
22 eqcom 2738 . . . . . . . . . . . 12 ((π‘“β€˜π‘) = (π‘“β€˜suc 𝑏) ↔ (π‘“β€˜suc 𝑏) = (π‘“β€˜π‘))
2321, 22orbi12i 912 . . . . . . . . . . 11 (((π‘“β€˜π‘)β—‘ [⊊] (π‘“β€˜suc 𝑏) ∨ (π‘“β€˜π‘) = (π‘“β€˜suc 𝑏)) ↔ ((π‘“β€˜suc 𝑏) ⊊ (π‘“β€˜π‘) ∨ (π‘“β€˜suc 𝑏) = (π‘“β€˜π‘)))
2416, 23sylbb2 237 . . . . . . . . . 10 ((π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘) β†’ ((π‘“β€˜π‘)β—‘ [⊊] (π‘“β€˜suc 𝑏) ∨ (π‘“β€˜π‘) = (π‘“β€˜suc 𝑏)))
2524ralimi 3082 . . . . . . . . 9 (βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘) β†’ βˆ€π‘ ∈ Ο‰ ((π‘“β€˜π‘)β—‘ [⊊] (π‘“β€˜suc 𝑏) ∨ (π‘“β€˜π‘) = (π‘“β€˜suc 𝑏)))
2625ad2antll 726 . . . . . . . 8 ((𝐴 ∈ FinII ∧ (𝑓:Ο‰βŸΆπ’« 𝐴 ∧ βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘))) β†’ βˆ€π‘ ∈ Ο‰ ((π‘“β€˜π‘)β—‘ [⊊] (π‘“β€˜suc 𝑏) ∨ (π‘“β€˜π‘) = (π‘“β€˜suc 𝑏)))
27 porpss 7721 . . . . . . . . . 10 [⊊] Po ran 𝑓
28 cnvpo 6286 . . . . . . . . . 10 ( [⊊] Po ran 𝑓 ↔ β—‘ [⊊] Po ran 𝑓)
2927, 28mpbi 229 . . . . . . . . 9 β—‘ [⊊] Po ran 𝑓
3029a1i 11 . . . . . . . 8 ((𝐴 ∈ FinII ∧ (𝑓:Ο‰βŸΆπ’« 𝐴 ∧ βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘))) β†’ β—‘ [⊊] Po ran 𝑓)
31 sornom 10276 . . . . . . . 8 ((𝑓 Fn Ο‰ ∧ βˆ€π‘ ∈ Ο‰ ((π‘“β€˜π‘)β—‘ [⊊] (π‘“β€˜suc 𝑏) ∨ (π‘“β€˜π‘) = (π‘“β€˜suc 𝑏)) ∧ β—‘ [⊊] Po ran 𝑓) β†’ β—‘ [⊊] Or ran 𝑓)
3215, 26, 30, 31syl3anc 1370 . . . . . . 7 ((𝐴 ∈ FinII ∧ (𝑓:Ο‰βŸΆπ’« 𝐴 ∧ βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘))) β†’ β—‘ [⊊] Or ran 𝑓)
33 cnvso 6287 . . . . . . 7 ( [⊊] Or ran 𝑓 ↔ β—‘ [⊊] Or ran 𝑓)
3432, 33sylibr 233 . . . . . 6 ((𝐴 ∈ FinII ∧ (𝑓:Ο‰βŸΆπ’« 𝐴 ∧ βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘))) β†’ [⊊] Or ran 𝑓)
35 fin2i2 10317 . . . . . 6 (((𝐴 ∈ FinII ∧ ran 𝑓 βŠ† 𝒫 𝐴) ∧ (ran 𝑓 β‰  βˆ… ∧ [⊊] Or ran 𝑓)) β†’ ∩ ran 𝑓 ∈ ran 𝑓)
362, 4, 13, 34, 35syl22anc 836 . . . . 5 ((𝐴 ∈ FinII ∧ (𝑓:Ο‰βŸΆπ’« 𝐴 ∧ βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘))) β†’ ∩ ran 𝑓 ∈ ran 𝑓)
3736expr 456 . . . 4 ((𝐴 ∈ FinII ∧ 𝑓:Ο‰βŸΆπ’« 𝐴) β†’ (βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘) β†’ ∩ ran 𝑓 ∈ ran 𝑓))
381, 37sylan2 592 . . 3 ((𝐴 ∈ FinII ∧ 𝑓 ∈ (𝒫 𝐴 ↑m Ο‰)) β†’ (βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘) β†’ ∩ ran 𝑓 ∈ ran 𝑓))
3938ralrimiva 3145 . 2 (𝐴 ∈ FinII β†’ βˆ€π‘“ ∈ (𝒫 𝐴 ↑m Ο‰)(βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘) β†’ ∩ ran 𝑓 ∈ ran 𝑓))
40 fin23lem40.f . . 3 𝐹 = {𝑔 ∣ βˆ€π‘Ž ∈ (𝒫 𝑔 ↑m Ο‰)(βˆ€π‘₯ ∈ Ο‰ (π‘Žβ€˜suc π‘₯) βŠ† (π‘Žβ€˜π‘₯) β†’ ∩ ran π‘Ž ∈ ran π‘Ž)}
4140isfin3ds 10328 . 2 (𝐴 ∈ FinII β†’ (𝐴 ∈ 𝐹 ↔ βˆ€π‘“ ∈ (𝒫 𝐴 ↑m Ο‰)(βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘) β†’ ∩ ran 𝑓 ∈ ran 𝑓)))
4239, 41mpbird 257 1 (𝐴 ∈ FinII β†’ 𝐴 ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∨ wo 844   = wceq 1540   ∈ wcel 2105  {cab 2708   β‰  wne 2939  βˆ€wral 3060   βŠ† wss 3948   ⊊ wpss 3949  βˆ…c0 4322  π’« cpw 4602  βˆ© cint 4950   class class class wbr 5148   Po wpo 5586   Or wor 5587  β—‘ccnv 5675  dom cdm 5676  ran crn 5677  suc csuc 6366   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412   [⊊] crpss 7716  Ο‰com 7859   ↑m cmap 8824  FinIIcfin2 10278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-rpss 7717  df-om 7860  df-1st 7979  df-2nd 7980  df-map 8826  df-fin2 10285
This theorem is referenced by:  fin23  10388
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