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Theorem fin23lem20 9119
Description: Lemma for fin23 9171. 𝑋 is either contained in or disjoint from all input sets. (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
fin23lem20 (𝐴 ∈ ω → ( ran 𝑈 ⊆ (𝑡𝐴) ∨ ( ran 𝑈 ∩ (𝑡𝐴)) = ∅))
Distinct variable groups:   𝑡,𝑖,𝑢   𝐴,𝑖,𝑢   𝑈,𝑖,𝑢
Allowed substitution hints:   𝐴(𝑡)   𝑈(𝑡)

Proof of Theorem fin23lem20
StepHypRef Expression
1 fin23lem.a . . . . 5 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
21fnseqom 7510 . . . 4 𝑈 Fn ω
3 peano2 7048 . . . 4 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
4 fnfvelrn 6322 . . . 4 ((𝑈 Fn ω ∧ suc 𝐴 ∈ ω) → (𝑈‘suc 𝐴) ∈ ran 𝑈)
52, 3, 4sylancr 694 . . 3 (𝐴 ∈ ω → (𝑈‘suc 𝐴) ∈ ran 𝑈)
6 intss1 4464 . . 3 ((𝑈‘suc 𝐴) ∈ ran 𝑈 ran 𝑈 ⊆ (𝑈‘suc 𝐴))
75, 6syl 17 . 2 (𝐴 ∈ ω → ran 𝑈 ⊆ (𝑈‘suc 𝐴))
81fin23lem19 9118 . 2 (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅))
9 sstr2 3595 . . 3 ( ran 𝑈 ⊆ (𝑈‘suc 𝐴) → ((𝑈‘suc 𝐴) ⊆ (𝑡𝐴) → ran 𝑈 ⊆ (𝑡𝐴)))
10 ssdisj 4004 . . . 4 (( ran 𝑈 ⊆ (𝑈‘suc 𝐴) ∧ ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅) → ( ran 𝑈 ∩ (𝑡𝐴)) = ∅)
1110ex 450 . . 3 ( ran 𝑈 ⊆ (𝑈‘suc 𝐴) → (((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅ → ( ran 𝑈 ∩ (𝑡𝐴)) = ∅))
129, 11orim12d 882 . 2 ( ran 𝑈 ⊆ (𝑈‘suc 𝐴) → (((𝑈‘suc 𝐴) ⊆ (𝑡𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅) → ( ran 𝑈 ⊆ (𝑡𝐴) ∨ ( ran 𝑈 ∩ (𝑡𝐴)) = ∅)))
137, 8, 12sylc 65 1 (𝐴 ∈ ω → ( ran 𝑈 ⊆ (𝑡𝐴) ∨ ( ran 𝑈 ∩ (𝑡𝐴)) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383   = wceq 1480  wcel 1987  Vcvv 3190  cin 3559  wss 3560  c0 3897  ifcif 4064   cuni 4409   cint 4447  ran crn 5085  suc csuc 5694   Fn wfn 5852  cfv 5857  cmpt2 6617  ωcom 7027  seq𝜔cseqom 7502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-seqom 7503
This theorem is referenced by:  fin23lem30  9124
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