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Theorem fin23lem33 9565
Description: Lemma for fin23 9609. 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 6499 . . . . . . 7 (𝑗 = 𝑐 → (𝑒𝑗) = (𝑒𝑐))
21ineq1d 4075 . . . . . 6 (𝑗 = 𝑐 → ((𝑒𝑗) ∩ 𝑘) = ((𝑒𝑐) ∩ 𝑘))
32eqeq1d 2780 . . . . 5 (𝑗 = 𝑐 → (((𝑒𝑗) ∩ 𝑘) = ∅ ↔ ((𝑒𝑐) ∩ 𝑘) = ∅))
43, 2ifbieq2d 4375 . . . 4 (𝑗 = 𝑐 → if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘)) = if(((𝑒𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑐) ∩ 𝑘)))
5 ineq2 4070 . . . . . 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 4377 . . . 4 (𝑘 = 𝑑 → if(((𝑒𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑐) ∩ 𝑘)) = if(((𝑒𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒𝑐) ∩ 𝑑)))
94, 8cbvmpov 7065 . . 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 7893 . . 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 679 . 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 6499 . . . 4 (𝑙 = 𝑦 → (𝑒𝑙) = (𝑒𝑦))
1514sseq2d 3889 . . 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 3412 . 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 9564 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 387  wal 1505   = wceq 1507  wex 1742  wcel 2050  {cab 2758  wral 3088  {crab 3092  Vcvv 3415  cdif 3826  cin 3828  wss 3829  wpss 3830  c0 4178  ifcif 4350  𝒫 cpw 4422   cuni 4712   cint 4749   class class class wbr 4929  cmpt 5008  ran crn 5408  ccom 5411  suc csuc 6031  1-1wf1 6185  cfv 6188  crio 6936  (class class class)co 6976  cmpo 6978  ωcom 7396  seq𝜔cseqom 7886  𝑚 cmap 8206  cen 8303  Fincfn 8306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-seqom 7887  df-1o 7905  df-oadd 7909  df-er 8089  df-map 8208  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-card 9162
This theorem is referenced by:  fin23lem41  9572
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