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Theorem fin23lem12 9138
Description: The beginning of the proof that every II-finite set (every chain of subsets has a maximal element) is III-finite (has no denumerable collection of subsets).

This first section is dedicated to the construction of 𝑈 and its intersection. First, the value of 𝑈 at a successor. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Hypothesis
Ref Expression
fin23lem.a 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
Assertion
Ref Expression
fin23lem12 (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
Distinct variable groups:   𝑡,𝑖,𝑢   𝐴,𝑖,𝑢   𝑈,𝑖,𝑢
Allowed substitution hints:   𝐴(𝑡)   𝑈(𝑡)

Proof of Theorem fin23lem12
StepHypRef Expression
1 fin23lem.a . . 3 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
21seqomsuc 7537 . 2 (𝐴 ∈ ω → (𝑈‘suc 𝐴) = (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)))(𝑈𝐴)))
3 fvex 6188 . . 3 (𝑈𝐴) ∈ V
4 fveq2 6178 . . . . . . 7 (𝑖 = 𝐴 → (𝑡𝑖) = (𝑡𝐴))
54ineq1d 3805 . . . . . 6 (𝑖 = 𝐴 → ((𝑡𝑖) ∩ 𝑢) = ((𝑡𝐴) ∩ 𝑢))
65eqeq1d 2622 . . . . 5 (𝑖 = 𝐴 → (((𝑡𝑖) ∩ 𝑢) = ∅ ↔ ((𝑡𝐴) ∩ 𝑢) = ∅))
76, 5ifbieq2d 4102 . . . 4 (𝑖 = 𝐴 → if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)) = if(((𝑡𝐴) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝐴) ∩ 𝑢)))
8 ineq2 3800 . . . . . 6 (𝑢 = (𝑈𝐴) → ((𝑡𝐴) ∩ 𝑢) = ((𝑡𝐴) ∩ (𝑈𝐴)))
98eqeq1d 2622 . . . . 5 (𝑢 = (𝑈𝐴) → (((𝑡𝐴) ∩ 𝑢) = ∅ ↔ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅))
10 id 22 . . . . 5 (𝑢 = (𝑈𝐴) → 𝑢 = (𝑈𝐴))
119, 10, 8ifbieq12d 4104 . . . 4 (𝑢 = (𝑈𝐴) → if(((𝑡𝐴) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝐴) ∩ 𝑢)) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
12 eqid 2620 . . . 4 (𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))) = (𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)))
133inex2 4791 . . . . 5 ((𝑡𝐴) ∩ (𝑈𝐴)) ∈ V
143, 13ifex 4147 . . . 4 if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))) ∈ V
157, 11, 12, 14ovmpt2 6781 . . 3 ((𝐴 ∈ ω ∧ (𝑈𝐴) ∈ V) → (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)))(𝑈𝐴)) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
163, 15mpan2 706 . 2 (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)))(𝑈𝐴)) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
172, 16eqtrd 2654 1 (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  Vcvv 3195  cin 3566  c0 3907  ifcif 4077   cuni 4427  ran crn 5105  suc csuc 5713  cfv 5876  (class class class)co 6635  cmpt2 6637  ωcom 7050  seq𝜔cseqom 7527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-seqom 7528
This theorem is referenced by:  fin23lem13  9139  fin23lem14  9140  fin23lem19  9143
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