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Theorem fin23lem33 10342
Description: Lemma for fin23 10386. 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 6885 . . . . . . 7 (𝑗 = 𝑐 → (𝑒𝑗) = (𝑒𝑐))
21ineq1d 4206 . . . . . 6 (𝑗 = 𝑐 → ((𝑒𝑗) ∩ 𝑘) = ((𝑒𝑐) ∩ 𝑘))
32eqeq1d 2728 . . . . 5 (𝑗 = 𝑐 → (((𝑒𝑗) ∩ 𝑘) = ∅ ↔ ((𝑒𝑐) ∩ 𝑘) = ∅))
43, 2ifbieq2d 4549 . . . 4 (𝑗 = 𝑐 → if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘)) = if(((𝑒𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑐) ∩ 𝑘)))
5 ineq2 4201 . . . . . 6 (𝑘 = 𝑑 → ((𝑒𝑐) ∩ 𝑘) = ((𝑒𝑐) ∩ 𝑑))
65eqeq1d 2728 . . . . 5 (𝑘 = 𝑑 → (((𝑒𝑐) ∩ 𝑘) = ∅ ↔ ((𝑒𝑐) ∩ 𝑑) = ∅))
7 id 22 . . . . 5 (𝑘 = 𝑑𝑘 = 𝑑)
86, 7, 5ifbieq12d 4551 . . . 4 (𝑘 = 𝑑 → if(((𝑒𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑐) ∩ 𝑘)) = if(((𝑒𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒𝑐) ∩ 𝑑)))
94, 8cbvmpov 7500 . . 3 (𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ if(((𝑒𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒𝑐) ∩ 𝑑)))
10 eqid 2726 . . 3 ran 𝑒 = ran 𝑒
11 seqomeq12 8455 . . 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 689 . 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 6885 . . . 4 (𝑙 = 𝑦 → (𝑒𝑙) = (𝑒𝑦))
1514sseq2d 4009 . . 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 3436 . 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 2726 . 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 2726 . 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 2726 . 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 10341 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 395  wal 1531   = wceq 1533  wex 1773  wcel 2098  {cab 2703  wral 3055  {crab 3426  Vcvv 3468  cdif 3940  cin 3942  wss 3943  wpss 3944  c0 4317  ifcif 4523  𝒫 cpw 4597   cuni 4902   cint 4943   class class class wbr 5141  cmpt 5224  ran crn 5670  ccom 5673  suc csuc 6360  1-1wf1 6534  cfv 6537  crio 7360  (class class class)co 7405  cmpo 7407  ωcom 7852  seqωcseqom 8448  m cmap 8822  cen 8938  Fincfn 8941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-seqom 8449  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936
This theorem is referenced by:  fin23lem41  10349
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