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Theorem fin23lem33 9127
 Description: Lemma for fin23 9171. Discharge hypotheses. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypothesis
Ref Expression
fin23lem33.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
Assertion
Ref Expression
fin23lem33 (𝐺𝐹 → ∃𝑓𝑏((𝑏:ω–1-1→V ∧ ran 𝑏𝐺) → ((𝑓𝑏):ω–1-1→V ∧ ran (𝑓𝑏) ⊊ ran 𝑏)))
Distinct variable groups:   𝑎,𝑏,𝑓,𝑔,𝑥,𝐺   𝐹,𝑎
Allowed substitution hints:   𝐹(𝑥,𝑓,𝑔,𝑏)

Proof of Theorem fin23lem33
Dummy variables 𝑐 𝑑 𝑒 𝑖 𝑗 𝑘 𝑙 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6158 . . . . . . 7 (𝑗 = 𝑐 → (𝑒𝑗) = (𝑒𝑐))
21ineq1d 3797 . . . . . 6 (𝑗 = 𝑐 → ((𝑒𝑗) ∩ 𝑘) = ((𝑒𝑐) ∩ 𝑘))
32eqeq1d 2623 . . . . 5 (𝑗 = 𝑐 → (((𝑒𝑗) ∩ 𝑘) = ∅ ↔ ((𝑒𝑐) ∩ 𝑘) = ∅))
43, 2ifbieq2d 4089 . . . 4 (𝑗 = 𝑐 → if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘)) = if(((𝑒𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑐) ∩ 𝑘)))
5 ineq2 3792 . . . . . 6 (𝑘 = 𝑑 → ((𝑒𝑐) ∩ 𝑘) = ((𝑒𝑐) ∩ 𝑑))
65eqeq1d 2623 . . . . 5 (𝑘 = 𝑑 → (((𝑒𝑐) ∩ 𝑘) = ∅ ↔ ((𝑒𝑐) ∩ 𝑑) = ∅))
7 id 22 . . . . 5 (𝑘 = 𝑑𝑘 = 𝑑)
86, 7, 5ifbieq12d 4091 . . . 4 (𝑘 = 𝑑 → if(((𝑒𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑐) ∩ 𝑘)) = if(((𝑒𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒𝑐) ∩ 𝑑)))
94, 8cbvmpt2v 6700 . . 3 (𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ if(((𝑒𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒𝑐) ∩ 𝑑)))
10 eqid 2621 . . 3 ran 𝑒 = ran 𝑒
11 seqomeq12 7509 . . 3 (((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ if(((𝑒𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒𝑐) ∩ 𝑑))) ∧ ran 𝑒 = ran 𝑒) → seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) = seq𝜔((𝑐 ∈ ω, 𝑑 ∈ V ↦ if(((𝑒𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒𝑐) ∩ 𝑑))), ran 𝑒))
129, 10, 11mp2an 707 . 2 seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) = seq𝜔((𝑐 ∈ ω, 𝑑 ∈ V ↦ if(((𝑒𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒𝑐) ∩ 𝑑))), ran 𝑒)
13 fin23lem33.f . 2 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
14 fveq2 6158 . . . 4 (𝑙 = 𝑦 → (𝑒𝑙) = (𝑒𝑦))
1514sseq2d 3618 . . 3 (𝑙 = 𝑦 → ( ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙) ↔ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑦)))
1615cbvrabv 3189 . 2 {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)} = {𝑦 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑦)}
17 eqid 2621 . 2 (𝑔 ∈ ω ↦ (𝑥 ∈ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)} (𝑥 ∩ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)}) ≈ 𝑔)) = (𝑔 ∈ ω ↦ (𝑥 ∈ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)} (𝑥 ∩ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)}) ≈ 𝑔))
18 eqid 2621 . 2 (𝑔 ∈ ω ↦ (𝑥 ∈ (ω ∖ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)})(𝑥 ∩ (ω ∖ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)})) ≈ 𝑔)) = (𝑔 ∈ ω ↦ (𝑥 ∈ (ω ∖ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)})(𝑥 ∩ (ω ∖ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)})) ≈ 𝑔))
19 eqid 2621 . 2 if({𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)} ∈ Fin, (𝑒 ∘ (𝑔 ∈ ω ↦ (𝑥 ∈ (ω ∖ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)})(𝑥 ∩ (ω ∖ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)})) ≈ 𝑔))), ((𝑖 ∈ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)} ↦ ((𝑒𝑖) ∖ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒))) ∘ (𝑔 ∈ ω ↦ (𝑥 ∈ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)} (𝑥 ∩ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)}) ≈ 𝑔)))) = if({𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)} ∈ Fin, (𝑒 ∘ (𝑔 ∈ ω ↦ (𝑥 ∈ (ω ∖ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)})(𝑥 ∩ (ω ∖ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)})) ≈ 𝑔))), ((𝑖 ∈ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)} ↦ ((𝑒𝑖) ∖ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒))) ∘ (𝑔 ∈ ω ↦ (𝑥 ∈ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)} (𝑥 ∩ {𝑙 ∈ ω ∣ ran seq𝜔((𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))), ran 𝑒) ⊆ (𝑒𝑙)}) ≈ 𝑔))))
2012, 13, 16, 17, 18, 19fin23lem32 9126 1 (𝐺𝐹 → ∃𝑓𝑏((𝑏:ω–1-1→V ∧ ran 𝑏𝐺) → ((𝑓𝑏):ω–1-1→V ∧ ran (𝑓𝑏) ⊊ ran 𝑏)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1478   = wceq 1480  ∃wex 1701   ∈ wcel 1987  {cab 2607  ∀wral 2908  {crab 2912  Vcvv 3190   ∖ cdif 3557   ∩ cin 3559   ⊆ wss 3560   ⊊ wpss 3561  ∅c0 3897  ifcif 4064  𝒫 cpw 4136  ∪ cuni 4409  ∩ cint 4447   class class class wbr 4623   ↦ cmpt 4683  ran crn 5085   ∘ ccom 5088  suc csuc 5694  –1-1→wf1 5854  ‘cfv 5857  ℩crio 6575  (class class class)co 6615   ↦ cmpt2 6617  ωcom 7027  seq𝜔cseqom 7502   ↑𝑚 cmap 7817   ≈ cen 7912  Fincfn 7915 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-seqom 7503  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725 This theorem is referenced by:  fin23lem41  9134
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