Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem20 Structured version   Visualization version   GIF version

Theorem fin23lem20 9360
 Description: Lemma for fin23 9412. 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 7702 . . . 4 𝑈 Fn ω
3 peano2 7232 . . . 4 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
4 fnfvelrn 6499 . . . 4 ((𝑈 Fn ω ∧ suc 𝐴 ∈ ω) → (𝑈‘suc 𝐴) ∈ ran 𝑈)
52, 3, 4sylancr 567 . . 3 (𝐴 ∈ ω → (𝑈‘suc 𝐴) ∈ ran 𝑈)
6 intss1 4624 . . 3 ((𝑈‘suc 𝐴) ∈ ran 𝑈 ran 𝑈 ⊆ (𝑈‘suc 𝐴))
75, 6syl 17 . 2 (𝐴 ∈ ω → ran 𝑈 ⊆ (𝑈‘suc 𝐴))
81fin23lem19 9359 . 2 (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅))
9 sstr2 3757 . . 3 ( ran 𝑈 ⊆ (𝑈‘suc 𝐴) → ((𝑈‘suc 𝐴) ⊆ (𝑡𝐴) → ran 𝑈 ⊆ (𝑡𝐴)))
10 ssdisj 4168 . . . 4 (( ran 𝑈 ⊆ (𝑈‘suc 𝐴) ∧ ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅) → ( ran 𝑈 ∩ (𝑡𝐴)) = ∅)
1110ex 397 . . 3 ( ran 𝑈 ⊆ (𝑈‘suc 𝐴) → (((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅ → ( ran 𝑈 ∩ (𝑡𝐴)) = ∅))
129, 11orim12d 945 . 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 826   = wceq 1630   ∈ wcel 2144  Vcvv 3349   ∩ cin 3720   ⊆ wss 3721  ∅c0 4061  ifcif 4223  ∪ cuni 4572  ∩ cint 4609  ran crn 5250  suc csuc 5868   Fn wfn 6026  ‘cfv 6031   ↦ cmpt2 6794  ωcom 7211  seq𝜔cseqom 7694 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  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 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-seqom 7695 This theorem is referenced by:  fin23lem30  9365
 Copyright terms: Public domain W3C validator