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Theorem fin23lem31 9203
Description: Lemma for fin23 9249. 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 9190 . . 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 9201 . . . . 5 ran 𝑍 ran 𝑡
98a1i 11 . . . 4 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑍 ran 𝑡)
103, 1fin23lem21 9199 . . . . . . 7 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ≠ ∅)
1110ancoms 468 . . . . . 6 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑈 ≠ ∅)
12 n0 3964 . . . . . 6 ( ran 𝑈 ≠ ∅ ↔ ∃𝑎 𝑎 ran 𝑈)
1311, 12sylib 208 . . . . 5 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ∃𝑎 𝑎 ran 𝑈)
143fnseqom 7595 . . . . . . . . . . . . . 14 𝑈 Fn ω
15 fndm 6028 . . . . . . . . . . . . . 14 (𝑈 Fn ω → dom 𝑈 = ω)
1614, 15ax-mp 5 . . . . . . . . . . . . 13 dom 𝑈 = ω
17 peano1 7127 . . . . . . . . . . . . . 14 ∅ ∈ ω
1817ne0ii 3956 . . . . . . . . . . . . 13 ω ≠ ∅
1916, 18eqnetri 2893 . . . . . . . . . . . 12 dom 𝑈 ≠ ∅
20 dm0rn0 5374 . . . . . . . . . . . . 13 (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
2120necon3bii 2875 . . . . . . . . . . . 12 (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
2219, 21mpbi 220 . . . . . . . . . . 11 ran 𝑈 ≠ ∅
23 intssuni 4531 . . . . . . . . . . 11 (ran 𝑈 ≠ ∅ → ran 𝑈 ran 𝑈)
2422, 23ax-mp 5 . . . . . . . . . 10 ran 𝑈 ran 𝑈
253fin23lem16 9195 . . . . . . . . . 10 ran 𝑈 = ran 𝑡
2624, 25sseqtri 3670 . . . . . . . . 9 ran 𝑈 ran 𝑡
2726sseli 3632 . . . . . . . 8 (𝑎 ran 𝑈𝑎 ran 𝑡)
2827adantl 481 . . . . . . 7 (((𝑡:ω–1-1𝑉 ran 𝑡𝐹) ∧ 𝑎 ran 𝑈) → 𝑎 ran 𝑡)
29 f1fun 6141 . . . . . . . . . . . . 13 (𝑡:ω–1-1𝑉 → Fun 𝑡)
3029adantr 480 . . . . . . . . . . . 12 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → Fun 𝑡)
313, 1, 4, 5, 6, 7fin23lem30 9202 . . . . . . . . . . . 12 (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)
3230, 31syl 17 . . . . . . . . . . 11 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ( ran 𝑍 ran 𝑈) = ∅)
33 disj 4050 . . . . . . . . . . 11 (( ran 𝑍 ran 𝑈) = ∅ ↔ ∀𝑎 ran 𝑍 ¬ 𝑎 ran 𝑈)
3432, 33sylib 208 . . . . . . . . . 10 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ∀𝑎 ran 𝑍 ¬ 𝑎 ran 𝑈)
35 rsp 2958 . . . . . . . . . 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 129 . . . . . . . 8 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → (𝑎 ran 𝑈 → ¬ 𝑎 ran 𝑍))
3837imp 444 . . . . . . 7 (((𝑡:ω–1-1𝑉 ran 𝑡𝐹) ∧ 𝑎 ran 𝑈) → ¬ 𝑎 ran 𝑍)
39 nelne1 2919 . . . . . . 7 ((𝑎 ran 𝑡 ∧ ¬ 𝑎 ran 𝑍) → ran 𝑡 ran 𝑍)
4028, 38, 39syl2anc 694 . . . . . 6 (((𝑡:ω–1-1𝑉 ran 𝑡𝐹) ∧ 𝑎 ran 𝑈) → ran 𝑡 ran 𝑍)
4140necomd 2878 . . . . 5 (((𝑡:ω–1-1𝑉 ran 𝑡𝐹) ∧ 𝑎 ran 𝑈) → ran 𝑍 ran 𝑡)
4213, 41exlimddv 1903 . . . 4 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑍 ran 𝑡)
43 df-pss 3623 . . . 4 ( ran 𝑍 ran 𝑡 ↔ ( ran 𝑍 ran 𝑡 ran 𝑍 ran 𝑡))
449, 42, 43sylanbrc 699 . . 3 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑍 ran 𝑡)
452, 44sylan2 490 . 2 ((𝑡:ω–1-1𝑉 ∧ (𝐺𝐹 ran 𝑡𝐺)) → ran 𝑍 ran 𝑡)
46453impb 1279 1 ((𝑡:ω–1-1𝑉𝐺𝐹 ran 𝑡𝐺) → ran 𝑍 ran 𝑡)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  {crab 2945  Vcvv 3231  cdif 3604  cin 3606  wss 3607  wpss 3608  c0 3948  ifcif 4119  𝒫 cpw 4191   cuni 4468   cint 4507   class class class wbr 4685  cmpt 4762  dom cdm 5143  ran crn 5144  ccom 5147  suc csuc 5763  Fun wfun 5920   Fn wfn 5921  1-1wf1 5923  cfv 5926  crio 6650  (class class class)co 6690  cmpt2 6692  ωcom 7107  seq𝜔cseqom 7587  𝑚 cmap 7899  cen 7994  Fincfn 7997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seqom 7588  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803
This theorem is referenced by:  fin23lem32  9204
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