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Theorem fin23lem17 9198
Description: Lemma for fin23 9249. 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 . . . . . 6 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
21fnseqom 7595 . . . . 5 𝑈 Fn ω
3 dffn3 6092 . . . . 5 (𝑈 Fn ω ↔ 𝑈:ω⟶ran 𝑈)
42, 3mpbi 220 . . . 4 𝑈:ω⟶ran 𝑈
5 pwuni 4506 . . . . 5 ran 𝑈 ⊆ 𝒫 ran 𝑈
61fin23lem16 9195 . . . . . 6 ran 𝑈 = ran 𝑡
76pweqi 4195 . . . . 5 𝒫 ran 𝑈 = 𝒫 ran 𝑡
85, 7sseqtri 3670 . . . 4 ran 𝑈 ⊆ 𝒫 ran 𝑡
9 fss 6094 . . . 4 ((𝑈:ω⟶ran 𝑈 ∧ ran 𝑈 ⊆ 𝒫 ran 𝑡) → 𝑈:ω⟶𝒫 ran 𝑡)
104, 8, 9mp2an 708 . . 3 𝑈:ω⟶𝒫 ran 𝑡
11 vex 3234 . . . . . . 7 𝑡 ∈ V
1211rnex 7142 . . . . . 6 ran 𝑡 ∈ V
1312uniex 6995 . . . . 5 ran 𝑡 ∈ V
1413pwex 4878 . . . 4 𝒫 ran 𝑡 ∈ V
15 f1f 6139 . . . . . 6 (𝑡:ω–1-1𝑉𝑡:ω⟶𝑉)
16 dmfex 7166 . . . . . 6 ((𝑡 ∈ V ∧ 𝑡:ω⟶𝑉) → ω ∈ V)
1711, 15, 16sylancr 696 . . . . 5 (𝑡:ω–1-1𝑉 → ω ∈ V)
1817adantl 481 . . . 4 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ω ∈ V)
19 elmapg 7912 . . . 4 ((𝒫 ran 𝑡 ∈ V ∧ ω ∈ V) → (𝑈 ∈ (𝒫 ran 𝑡𝑚 ω) ↔ 𝑈:ω⟶𝒫 ran 𝑡))
2014, 18, 19sylancr 696 . . 3 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → (𝑈 ∈ (𝒫 ran 𝑡𝑚 ω) ↔ 𝑈:ω⟶𝒫 ran 𝑡))
2110, 20mpbiri 248 . 2 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → 𝑈 ∈ (𝒫 ran 𝑡𝑚 ω))
22 fin23lem17.f . . . . 5 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
2322isfin3ds 9189 . . . 4 ( ran 𝑡𝐹 → ( ran 𝑡𝐹 ↔ ∀𝑏 ∈ (𝒫 ran 𝑡𝑚 ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏)))
2423ibi 256 . . 3 ( ran 𝑡𝐹 → ∀𝑏 ∈ (𝒫 ran 𝑡𝑚 ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏))
2524adantr 480 . 2 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ∀𝑏 ∈ (𝒫 ran 𝑡𝑚 ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏))
261fin23lem13 9192 . . . 4 (𝑐 ∈ ω → (𝑈‘suc 𝑐) ⊆ (𝑈𝑐))
2726rgen 2951 . . 3 𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)
2827a1i 11 . 2 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐))
29 fveq1 6228 . . . . . 6 (𝑏 = 𝑈 → (𝑏‘suc 𝑐) = (𝑈‘suc 𝑐))
30 fveq1 6228 . . . . . 6 (𝑏 = 𝑈 → (𝑏𝑐) = (𝑈𝑐))
3129, 30sseq12d 3667 . . . . 5 (𝑏 = 𝑈 → ((𝑏‘suc 𝑐) ⊆ (𝑏𝑐) ↔ (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)))
3231ralbidv 3015 . . . 4 (𝑏 = 𝑈 → (∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) ↔ ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)))
33 rneq 5383 . . . . . 6 (𝑏 = 𝑈 → ran 𝑏 = ran 𝑈)
3433inteqd 4512 . . . . 5 (𝑏 = 𝑈 ran 𝑏 = ran 𝑈)
3534, 33eleq12d 2724 . . . 4 (𝑏 = 𝑈 → ( ran 𝑏 ∈ ran 𝑏 ran 𝑈 ∈ ran 𝑈))
3632, 35imbi12d 333 . . 3 (𝑏 = 𝑈 → ((∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏) ↔ (∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐) → ran 𝑈 ∈ ran 𝑈)))
3736rspcv 3336 . 2 (𝑈 ∈ (𝒫 ran 𝑡𝑚 ω) → (∀𝑏 ∈ (𝒫 ran 𝑡𝑚 ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏) → (∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐) → ran 𝑈 ∈ ran 𝑈)))
3821, 25, 28, 37syl3c 66 1 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ran 𝑈 ∈ ran 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wral 2941  Vcvv 3231  cin 3606  wss 3607  c0 3948  ifcif 4119  𝒫 cpw 4191   cuni 4468   cint 4507  ran crn 5144  suc csuc 5763   Fn wfn 5921  wf 5922  1-1wf1 5923  cfv 5926  (class class class)co 6690  cmpt2 6692  ωcom 7107  seq𝜔cseqom 7587  𝑚 cmap 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seqom 7588  df-map 7901
This theorem is referenced by:  fin23lem21  9199
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