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Theorem fin23lem17 10378
Description: Lemma for fin23 10429. By ? Fin3DS ? , 𝑈 achieves its minimum (𝑋 in the synopsis above, but we will not be assigning a symbol here). TODO: Fix comment; math symbol Fin3DS does not exist. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Hypotheses
Ref Expression
fin23lem.a 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
fin23lem17.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
Assertion
Ref Expression
fin23lem17 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ∈ ran 𝑈)
Distinct variable groups:   𝑔,𝑖,𝑡,𝑢,𝑥,𝑎   𝐹,𝑎,𝑡   𝑉,𝑎   𝑥,𝑎   𝑈,𝑎,𝑖,𝑢   𝑔,𝑎
Allowed substitution hints:   𝑈(𝑥,𝑡,𝑔)   𝐹(𝑥,𝑢,𝑔,𝑖)   𝑉(𝑥,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem17
Dummy variables 𝑐 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fin23lem.a . . . 4 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
21fin23lem13 10372 . . 3 (𝑐 ∈ ω → (𝑈‘suc 𝑐) ⊆ (𝑈𝑐))
32rgen 3063 . 2 𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)
4 fveq1 6905 . . . . . 6 (𝑏 = 𝑈 → (𝑏‘suc 𝑐) = (𝑈‘suc 𝑐))
5 fveq1 6905 . . . . . 6 (𝑏 = 𝑈 → (𝑏𝑐) = (𝑈𝑐))
64, 5sseq12d 4017 . . . . 5 (𝑏 = 𝑈 → ((𝑏‘suc 𝑐) ⊆ (𝑏𝑐) ↔ (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)))
76ralbidv 3178 . . . 4 (𝑏 = 𝑈 → (∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) ↔ ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)))
8 rneq 5947 . . . . . 6 (𝑏 = 𝑈 → ran 𝑏 = ran 𝑈)
98inteqd 4951 . . . . 5 (𝑏 = 𝑈 ran 𝑏 = ran 𝑈)
109, 8eleq12d 2835 . . . 4 (𝑏 = 𝑈 → ( ran 𝑏 ∈ ran 𝑏 ran 𝑈 ∈ ran 𝑈))
117, 10imbi12d 344 . . 3 (𝑏 = 𝑈 → ((∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏) ↔ (∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐) → ran 𝑈 ∈ ran 𝑈)))
12 fin23lem17.f . . . . . 6 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
1312isfin3ds 10369 . . . . 5 ( ran 𝑡𝐹 → ( ran 𝑡𝐹 ↔ ∀𝑏 ∈ (𝒫 ran 𝑡m ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏)))
1413ibi 267 . . . 4 ( ran 𝑡𝐹 → ∀𝑏 ∈ (𝒫 ran 𝑡m ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏))
1514adantr 480 . . 3 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ∀𝑏 ∈ (𝒫 ran 𝑡m ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏))
161fnseqom 8495 . . . . . 6 𝑈 Fn ω
17 dffn3 6748 . . . . . 6 (𝑈 Fn ω ↔ 𝑈:ω⟶ran 𝑈)
1816, 17mpbi 230 . . . . 5 𝑈:ω⟶ran 𝑈
19 pwuni 4945 . . . . . 6 ran 𝑈 ⊆ 𝒫 ran 𝑈
201fin23lem16 10375 . . . . . . 7 ran 𝑈 = ran 𝑡
2120pweqi 4616 . . . . . 6 𝒫 ran 𝑈 = 𝒫 ran 𝑡
2219, 21sseqtri 4032 . . . . 5 ran 𝑈 ⊆ 𝒫 ran 𝑡
23 fss 6752 . . . . 5 ((𝑈:ω⟶ran 𝑈 ∧ ran 𝑈 ⊆ 𝒫 ran 𝑡) → 𝑈:ω⟶𝒫 ran 𝑡)
2418, 22, 23mp2an 692 . . . 4 𝑈:ω⟶𝒫 ran 𝑡
25 vex 3484 . . . . . . . 8 𝑡 ∈ V
2625rnex 7932 . . . . . . 7 ran 𝑡 ∈ V
2726uniex 7761 . . . . . 6 ran 𝑡 ∈ V
2827pwex 5380 . . . . 5 𝒫 ran 𝑡 ∈ V
29 f1f 6804 . . . . . . 7 (𝑡:ω–1-1𝑉𝑡:ω⟶𝑉)
30 dmfex 7927 . . . . . . 7 ((𝑡 ∈ V ∧ 𝑡:ω⟶𝑉) → ω ∈ V)
3125, 29, 30sylancr 587 . . . . . 6 (𝑡:ω–1-1𝑉 → ω ∈ V)
3231adantl 481 . . . . 5 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ω ∈ V)
33 elmapg 8879 . . . . 5 ((𝒫 ran 𝑡 ∈ V ∧ ω ∈ V) → (𝑈 ∈ (𝒫 ran 𝑡m ω) ↔ 𝑈:ω⟶𝒫 ran 𝑡))
3428, 32, 33sylancr 587 . . . 4 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → (𝑈 ∈ (𝒫 ran 𝑡m ω) ↔ 𝑈:ω⟶𝒫 ran 𝑡))
3524, 34mpbiri 258 . . 3 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → 𝑈 ∈ (𝒫 ran 𝑡m ω))
3611, 15, 35rspcdva 3623 . 2 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → (∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐) → ran 𝑈 ∈ ran 𝑈))
373, 36mpi 20 1 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ∈ ran 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  Vcvv 3480  cin 3950  wss 3951  c0 4333  ifcif 4525  𝒫 cpw 4600   cuni 4907   cint 4946  ran crn 5686  suc csuc 6386   Fn wfn 6556  wf 6557  1-1wf1 6558  cfv 6561  (class class class)co 7431  cmpo 7433  ωcom 7887  seqωcseqom 8487  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqom 8488  df-map 8868
This theorem is referenced by:  fin23lem21  10379
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