MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem12 Structured version   Visualization version   GIF version

Theorem fin23lem12 9355
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 7705 . 2 (𝐴 ∈ ω → (𝑈‘suc 𝐴) = (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)))(𝑈𝐴)))
3 fvex 6342 . . 3 (𝑈𝐴) ∈ V
4 fveq2 6332 . . . . . . 7 (𝑖 = 𝐴 → (𝑡𝑖) = (𝑡𝐴))
54ineq1d 3964 . . . . . 6 (𝑖 = 𝐴 → ((𝑡𝑖) ∩ 𝑢) = ((𝑡𝐴) ∩ 𝑢))
65eqeq1d 2773 . . . . 5 (𝑖 = 𝐴 → (((𝑡𝑖) ∩ 𝑢) = ∅ ↔ ((𝑡𝐴) ∩ 𝑢) = ∅))
76, 5ifbieq2d 4250 . . . 4 (𝑖 = 𝐴 → if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)) = if(((𝑡𝐴) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝐴) ∩ 𝑢)))
8 ineq2 3959 . . . . . 6 (𝑢 = (𝑈𝐴) → ((𝑡𝐴) ∩ 𝑢) = ((𝑡𝐴) ∩ (𝑈𝐴)))
98eqeq1d 2773 . . . . 5 (𝑢 = (𝑈𝐴) → (((𝑡𝐴) ∩ 𝑢) = ∅ ↔ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅))
10 id 22 . . . . 5 (𝑢 = (𝑈𝐴) → 𝑢 = (𝑈𝐴))
119, 10, 8ifbieq12d 4252 . . . 4 (𝑢 = (𝑈𝐴) → if(((𝑡𝐴) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝐴) ∩ 𝑢)) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
12 eqid 2771 . . . 4 (𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))) = (𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)))
133inex2 4934 . . . . 5 ((𝑡𝐴) ∩ (𝑈𝐴)) ∈ V
143, 13ifex 4295 . . . 4 if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))) ∈ V
157, 11, 12, 14ovmpt2 6943 . . 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 671 . 2 (𝐴 ∈ ω → (𝐴(𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢)))(𝑈𝐴)) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
172, 16eqtrd 2805 1 (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  Vcvv 3351  cin 3722  c0 4063  ifcif 4225   cuni 4574  ran crn 5250  suc csuc 5868  cfv 6031  (class class class)co 6793  cmpt2 6795  ωcom 7212  seq𝜔cseqom 7695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-seqom 7696
This theorem is referenced by:  fin23lem13  9356  fin23lem14  9357  fin23lem19  9360
  Copyright terms: Public domain W3C validator