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Theorem fin23lem33 10215
Description: Lemma for fin23 10259. 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 6838 . . . . . . 7 (𝑗 = 𝑐 → (𝑒𝑗) = (𝑒𝑐))
21ineq1d 4170 . . . . . 6 (𝑗 = 𝑐 → ((𝑒𝑗) ∩ 𝑘) = ((𝑒𝑐) ∩ 𝑘))
32eqeq1d 2740 . . . . 5 (𝑗 = 𝑐 → (((𝑒𝑗) ∩ 𝑘) = ∅ ↔ ((𝑒𝑐) ∩ 𝑘) = ∅))
43, 2ifbieq2d 4511 . . . 4 (𝑗 = 𝑐 → if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘)) = if(((𝑒𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑐) ∩ 𝑘)))
5 ineq2 4165 . . . . . 6 (𝑘 = 𝑑 → ((𝑒𝑐) ∩ 𝑘) = ((𝑒𝑐) ∩ 𝑑))
65eqeq1d 2740 . . . . 5 (𝑘 = 𝑑 → (((𝑒𝑐) ∩ 𝑘) = ∅ ↔ ((𝑒𝑐) ∩ 𝑑) = ∅))
7 id 22 . . . . 5 (𝑘 = 𝑑𝑘 = 𝑑)
86, 7, 5ifbieq12d 4513 . . . 4 (𝑘 = 𝑑 → if(((𝑒𝑐) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑐) ∩ 𝑘)) = if(((𝑒𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒𝑐) ∩ 𝑑)))
94, 8cbvmpov 7445 . . 3 (𝑗 ∈ ω, 𝑘 ∈ V ↦ if(((𝑒𝑗) ∩ 𝑘) = ∅, 𝑘, ((𝑒𝑗) ∩ 𝑘))) = (𝑐 ∈ ω, 𝑑 ∈ V ↦ if(((𝑒𝑐) ∩ 𝑑) = ∅, 𝑑, ((𝑒𝑐) ∩ 𝑑)))
10 eqid 2738 . . 3 ran 𝑒 = ran 𝑒
11 seqomeq12 8368 . . 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 691 . 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 6838 . . . 4 (𝑙 = 𝑦 → (𝑒𝑙) = (𝑒𝑦))
1514sseq2d 3975 . . 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 3416 . 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 2738 . 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 2738 . 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 2738 . 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 10214 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 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  {cab 2715  wral 3063  {crab 3406  Vcvv 3444  cdif 3906  cin 3908  wss 3909  wpss 3910  c0 4281  ifcif 4485  𝒫 cpw 4559   cuni 4864   cint 4906   class class class wbr 5104  cmpt 5187  ran crn 5632  ccom 5635  suc csuc 6316  1-1wf1 6489  cfv 6492  crio 7305  (class class class)co 7350  cmpo 7352  ωcom 7793  seqωcseqom 8361  m cmap 8699  cen 8814  Fincfn 8817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6250  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7306  df-ov 7353  df-oprab 7354  df-mpo 7355  df-om 7794  df-1st 7912  df-2nd 7913  df-frecs 8180  df-wrecs 8211  df-recs 8285  df-rdg 8324  df-seqom 8362  df-1o 8380  df-er 8582  df-map 8701  df-en 8818  df-dom 8819  df-sdom 8820  df-fin 8821  df-card 9809
This theorem is referenced by:  fin23lem41  10222
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