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Theorem fin23lem31 9022
Description: Lemma for fin23 9068. 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 9009 . . 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 9020 . . . . 5 ran 𝑍 ran 𝑡
98a1i 11 . . . 4 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑍 ran 𝑡)
103, 1fin23lem21 9018 . . . . . . 7 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ≠ ∅)
1110ancoms 467 . . . . . 6 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑈 ≠ ∅)
12 n0 3886 . . . . . 6 ( ran 𝑈 ≠ ∅ ↔ ∃𝑎 𝑎 ran 𝑈)
1311, 12sylib 206 . . . . 5 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ∃𝑎 𝑎 ran 𝑈)
143fnseqom 7411 . . . . . . . . . . . . . 14 𝑈 Fn ω
15 fndm 5887 . . . . . . . . . . . . . 14 (𝑈 Fn ω → dom 𝑈 = ω)
1614, 15ax-mp 5 . . . . . . . . . . . . 13 dom 𝑈 = ω
17 peano1 6951 . . . . . . . . . . . . . 14 ∅ ∈ ω
1817ne0ii 3878 . . . . . . . . . . . . 13 ω ≠ ∅
1916, 18eqnetri 2848 . . . . . . . . . . . 12 dom 𝑈 ≠ ∅
20 dm0rn0 5247 . . . . . . . . . . . . 13 (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
2120necon3bii 2830 . . . . . . . . . . . 12 (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
2219, 21mpbi 218 . . . . . . . . . . 11 ran 𝑈 ≠ ∅
23 intssuni 4425 . . . . . . . . . . 11 (ran 𝑈 ≠ ∅ → ran 𝑈 ran 𝑈)
2422, 23ax-mp 5 . . . . . . . . . 10 ran 𝑈 ran 𝑈
253fin23lem16 9014 . . . . . . . . . 10 ran 𝑈 = ran 𝑡
2624, 25sseqtri 3596 . . . . . . . . 9 ran 𝑈 ran 𝑡
2726sseli 3560 . . . . . . . 8 (𝑎 ran 𝑈𝑎 ran 𝑡)
2827adantl 480 . . . . . . 7 (((𝑡:ω–1-1𝑉 ran 𝑡𝐹) ∧ 𝑎 ran 𝑈) → 𝑎 ran 𝑡)
29 f1fun 5998 . . . . . . . . . . . . 13 (𝑡:ω–1-1𝑉 → Fun 𝑡)
3029adantr 479 . . . . . . . . . . . 12 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → Fun 𝑡)
313, 1, 4, 5, 6, 7fin23lem30 9021 . . . . . . . . . . . 12 (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)
3230, 31syl 17 . . . . . . . . . . 11 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ( ran 𝑍 ran 𝑈) = ∅)
33 disj 3965 . . . . . . . . . . 11 (( ran 𝑍 ran 𝑈) = ∅ ↔ ∀𝑎 ran 𝑍 ¬ 𝑎 ran 𝑈)
3432, 33sylib 206 . . . . . . . . . 10 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ∀𝑎 ran 𝑍 ¬ 𝑎 ran 𝑈)
35 rsp 2909 . . . . . . . . . 10 (∀𝑎 ran 𝑍 ¬ 𝑎 ran 𝑈 → (𝑎 ran 𝑍 → ¬ 𝑎 ran 𝑈))
3634, 35syl 17 . . . . . . . . 9 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → (𝑎 ran 𝑍 → ¬ 𝑎 ran 𝑈))
3736con2d 127 . . . . . . . 8 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → (𝑎 ran 𝑈 → ¬ 𝑎 ran 𝑍))
3837imp 443 . . . . . . 7 (((𝑡:ω–1-1𝑉 ran 𝑡𝐹) ∧ 𝑎 ran 𝑈) → ¬ 𝑎 ran 𝑍)
39 nelne1 2874 . . . . . . 7 ((𝑎 ran 𝑡 ∧ ¬ 𝑎 ran 𝑍) → ran 𝑡 ran 𝑍)
4028, 38, 39syl2anc 690 . . . . . 6 (((𝑡:ω–1-1𝑉 ran 𝑡𝐹) ∧ 𝑎 ran 𝑈) → ran 𝑡 ran 𝑍)
4140necomd 2833 . . . . 5 (((𝑡:ω–1-1𝑉 ran 𝑡𝐹) ∧ 𝑎 ran 𝑈) → ran 𝑍 ran 𝑡)
4213, 41exlimddv 1849 . . . 4 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑍 ran 𝑡)
43 df-pss 3552 . . . 4 ( ran 𝑍 ran 𝑡 ↔ ( ran 𝑍 ran 𝑡 ran 𝑍 ran 𝑡))
449, 42, 43sylanbrc 694 . . 3 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑍 ran 𝑡)
452, 44sylan2 489 . 2 ((𝑡:ω–1-1𝑉 ∧ (𝐺𝐹 ran 𝑡𝐺)) → ran 𝑍 ran 𝑡)
46453impb 1251 1 ((𝑡:ω–1-1𝑉𝐺𝐹 ran 𝑡𝐺) → ran 𝑍 ran 𝑡)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  {cab 2592  wne 2776  wral 2892  {crab 2896  Vcvv 3169  cdif 3533  cin 3535  wss 3536  wpss 3537  c0 3870  ifcif 4032  𝒫 cpw 4104   cuni 4363   cint 4401   class class class wbr 4574  cmpt 4634  dom cdm 5025  ran crn 5026  ccom 5029  suc csuc 5625  Fun wfun 5781   Fn wfn 5782  1-1wf1 5784  cfv 5787  crio 6485  (class class class)co 6524  cmpt2 6526  ωcom 6931  seq𝜔cseqom 7403  𝑚 cmap 7718  cen 7812  Fincfn 7815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-se 4985  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-isom 5796  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-seqom 7404  df-1o 7421  df-oadd 7425  df-er 7603  df-map 7720  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-card 8622
This theorem is referenced by:  fin23lem32  9023
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