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Theorem fin23lem12 9745
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 8087 . 2 (𝐴 ∈ ω → (𝑈‘suc 𝐴) = (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)))(𝑈𝐴)))
3 fvex 6679 . . 3 (𝑈𝐴) ∈ V
4 fveq2 6666 . . . . . . 7 (𝑖 = 𝐴 → (𝑡𝑖) = (𝑡𝐴))
54ineq1d 4191 . . . . . 6 (𝑖 = 𝐴 → ((𝑡𝑖) ∩ 𝑢) = ((𝑡𝐴) ∩ 𝑢))
65eqeq1d 2827 . . . . 5 (𝑖 = 𝐴 → (((𝑡𝑖) ∩ 𝑢) = ∅ ↔ ((𝑡𝐴) ∩ 𝑢) = ∅))
76, 5ifbieq2d 4494 . . . 4 (𝑖 = 𝐴 → if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)) = if(((𝑡𝐴) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝐴) ∩ 𝑢)))
8 ineq2 4186 . . . . . 6 (𝑢 = (𝑈𝐴) → ((𝑡𝐴) ∩ 𝑢) = ((𝑡𝐴) ∩ (𝑈𝐴)))
98eqeq1d 2827 . . . . 5 (𝑢 = (𝑈𝐴) → (((𝑡𝐴) ∩ 𝑢) = ∅ ↔ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅))
10 id 22 . . . . 5 (𝑢 = (𝑈𝐴) → 𝑢 = (𝑈𝐴))
119, 10, 8ifbieq12d 4496 . . . 4 (𝑢 = (𝑈𝐴) → if(((𝑡𝐴) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝐴) ∩ 𝑢)) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
12 eqid 2825 . . . 4 (𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))) = (𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)))
133inex2 5218 . . . . 5 ((𝑡𝐴) ∩ (𝑈𝐴)) ∈ V
143, 13ifex 4517 . . . 4 if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))) ∈ V
157, 11, 12, 14ovmpo 7303 . . 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 687 . 2 (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)))(𝑈𝐴)) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
172, 16eqtrd 2860 1 (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  Vcvv 3499  cin 3938  c0 4294  ifcif 4469   cuni 4836  ran crn 5554  suc csuc 6190  cfv 6351  (class class class)co 7151  cmpo 7153  ωcom 7571  seqωcseqom 8077
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-seqom 8078
This theorem is referenced by:  fin23lem13  9746  fin23lem14  9747  fin23lem19  9750
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