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Theorem fin23lem17 9758
Description: Lemma for fin23 9809. 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 9752 . . 3 (𝑐 ∈ ω → (𝑈‘suc 𝑐) ⊆ (𝑈𝑐))
32rgen 3143 . 2 𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)
4 fveq1 6660 . . . . . 6 (𝑏 = 𝑈 → (𝑏‘suc 𝑐) = (𝑈‘suc 𝑐))
5 fveq1 6660 . . . . . 6 (𝑏 = 𝑈 → (𝑏𝑐) = (𝑈𝑐))
64, 5sseq12d 3986 . . . . 5 (𝑏 = 𝑈 → ((𝑏‘suc 𝑐) ⊆ (𝑏𝑐) ↔ (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)))
76ralbidv 3192 . . . 4 (𝑏 = 𝑈 → (∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) ↔ ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)))
8 rneq 5793 . . . . . 6 (𝑏 = 𝑈 → ran 𝑏 = ran 𝑈)
98inteqd 4867 . . . . 5 (𝑏 = 𝑈 ran 𝑏 = ran 𝑈)
109, 8eleq12d 2910 . . . 4 (𝑏 = 𝑈 → ( ran 𝑏 ∈ ran 𝑏 ran 𝑈 ∈ ran 𝑈))
117, 10imbi12d 348 . . 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 9749 . . . . 5 ( ran 𝑡𝐹 → ( ran 𝑡𝐹 ↔ ∀𝑏 ∈ (𝒫 ran 𝑡m ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏)))
1413ibi 270 . . . 4 ( ran 𝑡𝐹 → ∀𝑏 ∈ (𝒫 ran 𝑡m ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏))
1514adantr 484 . . 3 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ∀𝑏 ∈ (𝒫 ran 𝑡m ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏))
161fnseqom 8087 . . . . . 6 𝑈 Fn ω
17 dffn3 6515 . . . . . 6 (𝑈 Fn ω ↔ 𝑈:ω⟶ran 𝑈)
1816, 17mpbi 233 . . . . 5 𝑈:ω⟶ran 𝑈
19 pwuni 4861 . . . . . 6 ran 𝑈 ⊆ 𝒫 ran 𝑈
201fin23lem16 9755 . . . . . . 7 ran 𝑈 = ran 𝑡
2120pweqi 4540 . . . . . 6 𝒫 ran 𝑈 = 𝒫 ran 𝑡
2219, 21sseqtri 3989 . . . . 5 ran 𝑈 ⊆ 𝒫 ran 𝑡
23 fss 6517 . . . . 5 ((𝑈:ω⟶ran 𝑈 ∧ ran 𝑈 ⊆ 𝒫 ran 𝑡) → 𝑈:ω⟶𝒫 ran 𝑡)
2418, 22, 23mp2an 691 . . . 4 𝑈:ω⟶𝒫 ran 𝑡
25 vex 3483 . . . . . . . 8 𝑡 ∈ V
2625rnex 7612 . . . . . . 7 ran 𝑡 ∈ V
2726uniex 7461 . . . . . 6 ran 𝑡 ∈ V
2827pwex 5268 . . . . 5 𝒫 ran 𝑡 ∈ V
29 f1f 6565 . . . . . . 7 (𝑡:ω–1-1𝑉𝑡:ω⟶𝑉)
30 dmfex 7636 . . . . . . 7 ((𝑡 ∈ V ∧ 𝑡:ω⟶𝑉) → ω ∈ V)
3125, 29, 30sylancr 590 . . . . . 6 (𝑡:ω–1-1𝑉 → ω ∈ V)
3231adantl 485 . . . . 5 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ω ∈ V)
33 elmapg 8415 . . . . 5 ((𝒫 ran 𝑡 ∈ V ∧ ω ∈ V) → (𝑈 ∈ (𝒫 ran 𝑡m ω) ↔ 𝑈:ω⟶𝒫 ran 𝑡))
3428, 32, 33sylancr 590 . . . 4 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → (𝑈 ∈ (𝒫 ran 𝑡m ω) ↔ 𝑈:ω⟶𝒫 ran 𝑡))
3524, 34mpbiri 261 . . 3 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → 𝑈 ∈ (𝒫 ran 𝑡m ω))
3611, 15, 35rspcdva 3611 . 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 209  wa 399   = wceq 1538  wcel 2115  {cab 2802  wral 3133  Vcvv 3480  cin 3918  wss 3919  c0 4276  ifcif 4450  𝒫 cpw 4522   cuni 4824   cint 4862  ran crn 5543  suc csuc 6180   Fn wfn 6338  wf 6339  1-1wf1 6340  cfv 6343  (class class class)co 7149  cmpo 7151  ωcom 7574  seqωcseqom 8079  m cmap 8402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-seqom 8080  df-map 8404
This theorem is referenced by:  fin23lem21  9759
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