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Theorem fin23lem33 9502
Description: Lemma for fin23 9546. 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 6446 . . . . . . 7 (𝑗 = 𝑐 → (𝑒𝑗) = (𝑒𝑐))
21ineq1d 4036 . . . . . 6 (𝑗 = 𝑐 → ((𝑒𝑗) ∩ 𝑘) = ((𝑒𝑐) ∩ 𝑘))
32eqeq1d 2780 . . . . 5 (𝑗 = 𝑐 → (((𝑒𝑗) ∩ 𝑘) = ∅ ↔ ((𝑒𝑐) ∩ 𝑘) = ∅))
43, 2ifbieq2d 4332 . . . 4 (𝑗 = 𝑐 → if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘)) = if(((𝑒𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑐) ∩ 𝑘)))
5 ineq2 4031 . . . . . 6 (𝑘 = 𝑑 → ((𝑒𝑐) ∩ 𝑘) = ((𝑒𝑐) ∩ 𝑑))
65eqeq1d 2780 . . . . 5 (𝑘 = 𝑑 → (((𝑒𝑐) ∩ 𝑘) = ∅ ↔ ((𝑒𝑐) ∩ 𝑑) = ∅))
7 id 22 . . . . 5 (𝑘 = 𝑑𝑘 = 𝑑)
86, 7, 5ifbieq12d 4334 . . . 4 (𝑘 = 𝑑 → if(((𝑒𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑐) ∩ 𝑘)) = if(((𝑒𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒𝑐) ∩ 𝑑)))
94, 8cbvmpt2v 7012 . . 3 (𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ if(((𝑒𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒𝑐) ∩ 𝑑)))
10 eqid 2778 . . 3 ran 𝑒 = ran 𝑒
11 seqomeq12 7832 . . 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 682 . 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 6446 . . . 4 (𝑙 = 𝑦 → (𝑒𝑙) = (𝑒𝑦))
1514sseq2d 3852 . . 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 3396 . 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 2778 . 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 2778 . 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 2778 . 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 9501 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 386  wal 1599   = wceq 1601  wex 1823  wcel 2107  {cab 2763  wral 3090  {crab 3094  Vcvv 3398  cdif 3789  cin 3791  wss 3792  wpss 3793  c0 4141  ifcif 4307  𝒫 cpw 4379   cuni 4671   cint 4710   class class class wbr 4886  cmpt 4965  ran crn 5356  ccom 5359  suc csuc 5978  1-1wf1 6132  cfv 6135  crio 6882  (class class class)co 6922  cmpt2 6924  ωcom 7343  seq𝜔cseqom 7825  𝑚 cmap 8140  cen 8238  Fincfn 8241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-seqom 7826  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098
This theorem is referenced by:  fin23lem41  9509
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