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Theorem fin23lem31 10383
Description: Lemma for fin23 10429. The residual is has a strictly smaller range than the previous sequence. This will be iterated to build an unbounded chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypotheses
Ref Expression
fin23lem.a 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
fin23lem17.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
fin23lem.b 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
fin23lem.c 𝑄 = (𝑤 ∈ ω ↦ (𝑥𝑃 (𝑥𝑃) ≈ 𝑤))
fin23lem.d 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
fin23lem.e 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
Assertion
Ref Expression
fin23lem31 ((𝑡:ω–1-1𝑉𝐺𝐹 ran 𝑡𝐺) → ran 𝑍 ran 𝑡)
Distinct variable groups:   𝑔,𝑖,𝑡,𝑢,𝑣,𝑥,𝑧,𝑎   𝐹,𝑎,𝑡   𝑉,𝑎   𝑤,𝑎,𝑥,𝑧,𝑃   𝑣,𝑎,𝑅,𝑖,𝑢   𝑈,𝑎,𝑖,𝑢,𝑣,𝑧   𝑍,𝑎   𝑔,𝑎,𝐺,𝑡,𝑥
Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑔,𝑖)   𝑄(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖,𝑎)   𝑅(𝑥,𝑧,𝑤,𝑡,𝑔)   𝑈(𝑥,𝑤,𝑡,𝑔)   𝐹(𝑥,𝑧,𝑤,𝑣,𝑢,𝑔,𝑖)   𝐺(𝑧,𝑤,𝑣,𝑢,𝑖)   𝑉(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)   𝑍(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem31
StepHypRef Expression
1 fin23lem17.f . . . 4 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
21ssfin3ds 10370 . . 3 ((𝐺𝐹 ran 𝑡𝐺) → ran 𝑡𝐹)
3 fin23lem.a . . . . . 6 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
4 fin23lem.b . . . . . 6 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
5 fin23lem.c . . . . . 6 𝑄 = (𝑤 ∈ ω ↦ (𝑥𝑃 (𝑥𝑃) ≈ 𝑤))
6 fin23lem.d . . . . . 6 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
7 fin23lem.e . . . . . 6 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
83, 1, 4, 5, 6, 7fin23lem29 10381 . . . . 5 ran 𝑍 ran 𝑡
98a1i 11 . . . 4 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑍 ran 𝑡)
103, 1fin23lem21 10379 . . . . . . 7 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ≠ ∅)
1110ancoms 458 . . . . . 6 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑈 ≠ ∅)
12 n0 4353 . . . . . 6 ( ran 𝑈 ≠ ∅ ↔ ∃𝑎 𝑎 ran 𝑈)
1311, 12sylib 218 . . . . 5 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ∃𝑎 𝑎 ran 𝑈)
143fnseqom 8495 . . . . . . . . . . . . 13 𝑈 Fn ω
15 fndm 6671 . . . . . . . . . . . . 13 (𝑈 Fn ω → dom 𝑈 = ω)
1614, 15ax-mp 5 . . . . . . . . . . . 12 dom 𝑈 = ω
17 peano1 7910 . . . . . . . . . . . . 13 ∅ ∈ ω
1817ne0ii 4344 . . . . . . . . . . . 12 ω ≠ ∅
1916, 18eqnetri 3011 . . . . . . . . . . 11 dom 𝑈 ≠ ∅
20 dm0rn0 5935 . . . . . . . . . . . 12 (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
2120necon3bii 2993 . . . . . . . . . . 11 (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
2219, 21mpbi 230 . . . . . . . . . 10 ran 𝑈 ≠ ∅
23 intssuni 4970 . . . . . . . . . 10 (ran 𝑈 ≠ ∅ → ran 𝑈 ran 𝑈)
2422, 23ax-mp 5 . . . . . . . . 9 ran 𝑈 ran 𝑈
253fin23lem16 10375 . . . . . . . . 9 ran 𝑈 = ran 𝑡
2624, 25sseqtri 4032 . . . . . . . 8 ran 𝑈 ran 𝑡
2726sseli 3979 . . . . . . 7 (𝑎 ran 𝑈𝑎 ran 𝑡)
28 f1fun 6806 . . . . . . . . . . . . 13 (𝑡:ω–1-1𝑉 → Fun 𝑡)
2928adantr 480 . . . . . . . . . . . 12 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → Fun 𝑡)
303, 1, 4, 5, 6, 7fin23lem30 10382 . . . . . . . . . . . 12 (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)
3129, 30syl 17 . . . . . . . . . . 11 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ( ran 𝑍 ran 𝑈) = ∅)
32 disj 4450 . . . . . . . . . . 11 (( ran 𝑍 ran 𝑈) = ∅ ↔ ∀𝑎 ran 𝑍 ¬ 𝑎 ran 𝑈)
3331, 32sylib 218 . . . . . . . . . 10 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ∀𝑎 ran 𝑍 ¬ 𝑎 ran 𝑈)
34 rsp 3247 . . . . . . . . . 10 (∀𝑎 ran 𝑍 ¬ 𝑎 ran 𝑈 → (𝑎 ran 𝑍 → ¬ 𝑎 ran 𝑈))
3533, 34syl 17 . . . . . . . . 9 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → (𝑎 ran 𝑍 → ¬ 𝑎 ran 𝑈))
3635con2d 134 . . . . . . . 8 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → (𝑎 ran 𝑈 → ¬ 𝑎 ran 𝑍))
3736imp 406 . . . . . . 7 (((𝑡:ω–1-1𝑉 ran 𝑡𝐹) ∧ 𝑎 ran 𝑈) → ¬ 𝑎 ran 𝑍)
38 nelne1 3039 . . . . . . 7 ((𝑎 ran 𝑡 ∧ ¬ 𝑎 ran 𝑍) → ran 𝑡 ran 𝑍)
3927, 37, 38syl2an2 686 . . . . . 6 (((𝑡:ω–1-1𝑉 ran 𝑡𝐹) ∧ 𝑎 ran 𝑈) → ran 𝑡 ran 𝑍)
4039necomd 2996 . . . . 5 (((𝑡:ω–1-1𝑉 ran 𝑡𝐹) ∧ 𝑎 ran 𝑈) → ran 𝑍 ran 𝑡)
4113, 40exlimddv 1935 . . . 4 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑍 ran 𝑡)
42 df-pss 3971 . . . 4 ( ran 𝑍 ran 𝑡 ↔ ( ran 𝑍 ran 𝑡 ran 𝑍 ran 𝑡))
439, 41, 42sylanbrc 583 . . 3 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑍 ran 𝑡)
442, 43sylan2 593 . 2 ((𝑡:ω–1-1𝑉 ∧ (𝐺𝐹 ran 𝑡𝐺)) → ran 𝑍 ran 𝑡)
45443impb 1115 1 ((𝑡:ω–1-1𝑉𝐺𝐹 ran 𝑡𝐺) → ran 𝑍 ran 𝑡)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  wral 3061  {crab 3436  Vcvv 3480  cdif 3948  cin 3950  wss 3951  wpss 3952  c0 4333  ifcif 4525  𝒫 cpw 4600   cuni 4907   cint 4946   class class class wbr 5143  cmpt 5225  dom cdm 5685  ran crn 5686  ccom 5689  suc csuc 6386  Fun wfun 6555   Fn wfn 6556  1-1wf1 6558  cfv 6561  crio 7387  (class class class)co 7431  cmpo 7433  ωcom 7887  seqωcseqom 8487  m cmap 8866  cen 8982  Fincfn 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqom 8488  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979
This theorem is referenced by:  fin23lem32  10384
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