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Theorem fin23lem12 9036
 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 7439 . 2 (𝐴 ∈ ω → (𝑈‘suc 𝐴) = (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)))(𝑈𝐴)))
3 fvex 6113 . . 3 (𝑈𝐴) ∈ V
4 fveq2 6103 . . . . . . 7 (𝑖 = 𝐴 → (𝑡𝑖) = (𝑡𝐴))
54ineq1d 3775 . . . . . 6 (𝑖 = 𝐴 → ((𝑡𝑖) ∩ 𝑢) = ((𝑡𝐴) ∩ 𝑢))
65eqeq1d 2612 . . . . 5 (𝑖 = 𝐴 → (((𝑡𝑖) ∩ 𝑢) = ∅ ↔ ((𝑡𝐴) ∩ 𝑢) = ∅))
76, 5ifbieq2d 4061 . . . 4 (𝑖 = 𝐴 → if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)) = if(((𝑡𝐴) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝐴) ∩ 𝑢)))
8 ineq2 3770 . . . . . 6 (𝑢 = (𝑈𝐴) → ((𝑡𝐴) ∩ 𝑢) = ((𝑡𝐴) ∩ (𝑈𝐴)))
98eqeq1d 2612 . . . . 5 (𝑢 = (𝑈𝐴) → (((𝑡𝐴) ∩ 𝑢) = ∅ ↔ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅))
10 id 22 . . . . 5 (𝑢 = (𝑈𝐴) → 𝑢 = (𝑈𝐴))
119, 10, 8ifbieq12d 4063 . . . 4 (𝑢 = (𝑈𝐴) → if(((𝑡𝐴) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝐴) ∩ 𝑢)) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
12 eqid 2610 . . . 4 (𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))) = (𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)))
133inex2 4728 . . . . 5 ((𝑡𝐴) ∩ (𝑈𝐴)) ∈ V
143, 13ifex 4106 . . . 4 if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))) ∈ V
157, 11, 12, 14ovmpt2 6694 . . 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 703 . 2 (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)))(𝑈𝐴)) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
172, 16eqtrd 2644 1 (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  ∅c0 3874  ifcif 4036  ∪ cuni 4372  ran crn 5039  suc csuc 5642  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957  seq𝜔cseqom 7429 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430 This theorem is referenced by:  fin23lem13  9037  fin23lem14  9038  fin23lem19  9041
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