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Theorem fin23lem31 10381
Description: Lemma for fin23 10427. 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 10368 . . 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 10379 . . . . 5 ran 𝑍 ran 𝑡
98a1i 11 . . . 4 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑍 ran 𝑡)
103, 1fin23lem21 10377 . . . . . . 7 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ≠ ∅)
1110ancoms 458 . . . . . 6 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑈 ≠ ∅)
12 n0 4359 . . . . . 6 ( ran 𝑈 ≠ ∅ ↔ ∃𝑎 𝑎 ran 𝑈)
1311, 12sylib 218 . . . . 5 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ∃𝑎 𝑎 ran 𝑈)
143fnseqom 8494 . . . . . . . . . . . . 13 𝑈 Fn ω
15 fndm 6672 . . . . . . . . . . . . 13 (𝑈 Fn ω → dom 𝑈 = ω)
1614, 15ax-mp 5 . . . . . . . . . . . 12 dom 𝑈 = ω
17 peano1 7911 . . . . . . . . . . . . 13 ∅ ∈ ω
1817ne0ii 4350 . . . . . . . . . . . 12 ω ≠ ∅
1916, 18eqnetri 3009 . . . . . . . . . . 11 dom 𝑈 ≠ ∅
20 dm0rn0 5938 . . . . . . . . . . . 12 (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
2120necon3bii 2991 . . . . . . . . . . 11 (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
2219, 21mpbi 230 . . . . . . . . . 10 ran 𝑈 ≠ ∅
23 intssuni 4975 . . . . . . . . . 10 (ran 𝑈 ≠ ∅ → ran 𝑈 ran 𝑈)
2422, 23ax-mp 5 . . . . . . . . 9 ran 𝑈 ran 𝑈
253fin23lem16 10373 . . . . . . . . 9 ran 𝑈 = ran 𝑡
2624, 25sseqtri 4032 . . . . . . . 8 ran 𝑈 ran 𝑡
2726sseli 3991 . . . . . . 7 (𝑎 ran 𝑈𝑎 ran 𝑡)
28 f1fun 6807 . . . . . . . . . . . . 13 (𝑡:ω–1-1𝑉 → Fun 𝑡)
2928adantr 480 . . . . . . . . . . . 12 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → Fun 𝑡)
303, 1, 4, 5, 6, 7fin23lem30 10380 . . . . . . . . . . . 12 (Fun 𝑡 → ( ran 𝑍 ran 𝑈) = ∅)
3129, 30syl 17 . . . . . . . . . . 11 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ( ran 𝑍 ran 𝑈) = ∅)
32 disj 4456 . . . . . . . . . . 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 3245 . . . . . . . . . 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 3037 . . . . . . 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 2994 . . . . 5 (((𝑡:ω–1-1𝑉 ran 𝑡𝐹) ∧ 𝑎 ran 𝑈) → ran 𝑍 ran 𝑡)
4113, 40exlimddv 1933 . . . 4 ((𝑡:ω–1-1𝑉 ran 𝑡𝐹) → ran 𝑍 ran 𝑡)
42 df-pss 3983 . . . 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 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  wral 3059  {crab 3433  Vcvv 3478  cdif 3960  cin 3962  wss 3963  wpss 3964  c0 4339  ifcif 4531  𝒫 cpw 4605   cuni 4912   cint 4951   class class class wbr 5148  cmpt 5231  dom cdm 5689  ran crn 5690  ccom 5693  suc csuc 6388  Fun wfun 6557   Fn wfn 6558  1-1wf1 6560  cfv 6563  crio 7387  (class class class)co 7431  cmpo 7433  ωcom 7887  seqωcseqom 8486  m cmap 8865  cen 8981  Fincfn 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977
This theorem is referenced by:  fin23lem32  10382
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