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Theorem fin23lem31 10087
Description: Lemma for fin23 10133. 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 10074 . . 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 10085 . . . . 5 ran 𝑍 ran 𝑡
98a1i 11 . . . 4 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑍 ran 𝑡)
103, 1fin23lem21 10083 . . . . . . 7 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ≠ ∅)
1110ancoms 459 . . . . . 6 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑈 ≠ ∅)
12 n0 4281 . . . . . 6 ( ran 𝑈 ≠ ∅ ↔ ∃𝑎 𝑎 ran 𝑈)
1311, 12sylib 217 . . . . 5 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ∃𝑎 𝑎 ran 𝑈)
143fnseqom 8274 . . . . . . . . . . . . 13 𝑈 Fn ω
15 fndm 6529 . . . . . . . . . . . . 13 (𝑈 Fn ω → dom 𝑈 = ω)
1614, 15ax-mp 5 . . . . . . . . . . . 12 dom 𝑈 = ω
17 peano1 7726 . . . . . . . . . . . . 13 ∅ ∈ ω
1817ne0ii 4272 . . . . . . . . . . . 12 ω ≠ ∅
1916, 18eqnetri 3014 . . . . . . . . . . 11 dom 𝑈 ≠ ∅
20 dm0rn0 5828 . . . . . . . . . . . 12 (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
2120necon3bii 2996 . . . . . . . . . . 11 (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
2219, 21mpbi 229 . . . . . . . . . 10 ran 𝑈 ≠ ∅
23 intssuni 4902 . . . . . . . . . 10 (ran 𝑈 ≠ ∅ → ran 𝑈 ran 𝑈)
2422, 23ax-mp 5 . . . . . . . . 9 ran 𝑈 ran 𝑈
253fin23lem16 10079 . . . . . . . . 9 ran 𝑈 = ran 𝑡
2624, 25sseqtri 3957 . . . . . . . 8 ran 𝑈 ran 𝑡
2726sseli 3917 . . . . . . 7 (𝑎 ran 𝑈𝑎 ran 𝑡)
28 f1fun 6665 . . . . . . . . . . . . 13 (𝑡:ω–1-1𝑉 → Fun 𝑡)
2928adantr 481 . . . . . . . . . . . 12 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → Fun 𝑡)
303, 1, 4, 5, 6, 7fin23lem30 10086 . . . . . . . . . . . 12 (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)
3129, 30syl 17 . . . . . . . . . . 11 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ( ran 𝑍 ran 𝑈) = ∅)
32 disj 4382 . . . . . . . . . . 11 (( ran 𝑍 ran 𝑈) = ∅ ↔ ∀𝑎 ran 𝑍 ¬ 𝑎 ran 𝑈)
3331, 32sylib 217 . . . . . . . . . 10 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ∀𝑎 ran 𝑍 ¬ 𝑎 ran 𝑈)
34 rsp 3130 . . . . . . . . . 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 407 . . . . . . 7 (((𝑡:ω–1-1𝑉 ran 𝑡𝐹) ∧ 𝑎 ran 𝑈) → ¬ 𝑎 ran 𝑍)
38 nelne1 3041 . . . . . . 7 ((𝑎 ran 𝑡 ∧ ¬ 𝑎 ran 𝑍) → ran 𝑡 ran 𝑍)
3927, 37, 38syl2an2 683 . . . . . 6 (((𝑡:ω–1-1𝑉 ran 𝑡𝐹) ∧ 𝑎 ran 𝑈) → ran 𝑡 ran 𝑍)
4039necomd 2999 . . . . 5 (((𝑡:ω–1-1𝑉 ran 𝑡𝐹) ∧ 𝑎 ran 𝑈) → ran 𝑍 ran 𝑡)
4113, 40exlimddv 1938 . . . 4 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑍 ran 𝑡)
42 df-pss 3906 . . . 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 1114 1 ((𝑡:ω–1-1𝑉𝐺𝐹 ran 𝑡𝐺) → ran 𝑍 ran 𝑡)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1086   = wceq 1539  wex 1782  wcel 2106  {cab 2715  wne 2943  wral 3064  {crab 3068  Vcvv 3430  cdif 3884  cin 3886  wss 3887  wpss 3888  c0 4257  ifcif 4460  𝒫 cpw 4534   cuni 4840   cint 4880   class class class wbr 5074  cmpt 5157  dom cdm 5585  ran crn 5586  ccom 5589  suc csuc 6262  Fun wfun 6421   Fn wfn 6422  1-1wf1 6424  cfv 6427  crio 7224  (class class class)co 7268  cmpo 7270  ωcom 7703  seqωcseqom 8266  m cmap 8603  cen 8718  Fincfn 8721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5222  ax-nul 5229  ax-pow 5287  ax-pr 5351  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-int 4881  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5485  df-eprel 5491  df-po 5499  df-so 5500  df-fr 5540  df-se 5541  df-we 5542  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-pred 6196  df-ord 6263  df-on 6264  df-lim 6265  df-suc 6266  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-f1 6432  df-fo 6433  df-f1o 6434  df-fv 6435  df-isom 6436  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7704  df-1st 7821  df-2nd 7822  df-frecs 8085  df-wrecs 8116  df-recs 8190  df-rdg 8229  df-seqom 8267  df-1o 8285  df-er 8486  df-map 8605  df-en 8722  df-dom 8723  df-sdom 8724  df-fin 8725  df-card 9685
This theorem is referenced by:  fin23lem32  10088
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