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Theorem fin23lem17 9372
Description: Lemma for fin23 9423. 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 . . . . 5 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
21fin23lem13 9366 . . . 4 (𝑐 ∈ ω → (𝑈‘suc 𝑐) ⊆ (𝑈𝑐))
32rgen 3060 . . 3 𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)
43a1i 11 . 2 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐))
5 fveq1 6352 . . . . . 6 (𝑏 = 𝑈 → (𝑏‘suc 𝑐) = (𝑈‘suc 𝑐))
6 fveq1 6352 . . . . . 6 (𝑏 = 𝑈 → (𝑏𝑐) = (𝑈𝑐))
75, 6sseq12d 3775 . . . . 5 (𝑏 = 𝑈 → ((𝑏‘suc 𝑐) ⊆ (𝑏𝑐) ↔ (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)))
87ralbidv 3124 . . . 4 (𝑏 = 𝑈 → (∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) ↔ ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)))
9 rneq 5506 . . . . . 6 (𝑏 = 𝑈 → ran 𝑏 = ran 𝑈)
109inteqd 4632 . . . . 5 (𝑏 = 𝑈 ran 𝑏 = ran 𝑈)
1110, 9eleq12d 2833 . . . 4 (𝑏 = 𝑈 → ( ran 𝑏 ∈ ran 𝑏 ran 𝑈 ∈ ran 𝑈))
128, 11imbi12d 333 . . 3 (𝑏 = 𝑈 → ((∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏) ↔ (∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐) → ran 𝑈 ∈ ran 𝑈)))
13 fin23lem17.f . . . . . 6 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
1413isfin3ds 9363 . . . . 5 ( ran 𝑡𝐹 → ( ran 𝑡𝐹 ↔ ∀𝑏 ∈ (𝒫 ran 𝑡𝑚 ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏)))
1514ibi 256 . . . 4 ( ran 𝑡𝐹 → ∀𝑏 ∈ (𝒫 ran 𝑡𝑚 ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏))
1615adantr 472 . . 3 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ∀𝑏 ∈ (𝒫 ran 𝑡𝑚 ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏))
171fnseqom 7720 . . . . . 6 𝑈 Fn ω
18 dffn3 6215 . . . . . 6 (𝑈 Fn ω ↔ 𝑈:ω⟶ran 𝑈)
1917, 18mpbi 220 . . . . 5 𝑈:ω⟶ran 𝑈
20 pwuni 4626 . . . . . 6 ran 𝑈 ⊆ 𝒫 ran 𝑈
211fin23lem16 9369 . . . . . . 7 ran 𝑈 = ran 𝑡
2221pweqi 4306 . . . . . 6 𝒫 ran 𝑈 = 𝒫 ran 𝑡
2320, 22sseqtri 3778 . . . . 5 ran 𝑈 ⊆ 𝒫 ran 𝑡
24 fss 6217 . . . . 5 ((𝑈:ω⟶ran 𝑈 ∧ ran 𝑈 ⊆ 𝒫 ran 𝑡) → 𝑈:ω⟶𝒫 ran 𝑡)
2519, 23, 24mp2an 710 . . . 4 𝑈:ω⟶𝒫 ran 𝑡
26 vex 3343 . . . . . . . 8 𝑡 ∈ V
2726rnex 7266 . . . . . . 7 ran 𝑡 ∈ V
2827uniex 7119 . . . . . 6 ran 𝑡 ∈ V
2928pwex 4997 . . . . 5 𝒫 ran 𝑡 ∈ V
30 f1f 6262 . . . . . . 7 (𝑡:ω–1-1𝑉𝑡:ω⟶𝑉)
31 dmfex 7290 . . . . . . 7 ((𝑡 ∈ V ∧ 𝑡:ω⟶𝑉) → ω ∈ V)
3226, 30, 31sylancr 698 . . . . . 6 (𝑡:ω–1-1𝑉 → ω ∈ V)
3332adantl 473 . . . . 5 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ω ∈ V)
34 elmapg 8038 . . . . 5 ((𝒫 ran 𝑡 ∈ V ∧ ω ∈ V) → (𝑈 ∈ (𝒫 ran 𝑡𝑚 ω) ↔ 𝑈:ω⟶𝒫 ran 𝑡))
3529, 33, 34sylancr 698 . . . 4 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → (𝑈 ∈ (𝒫 ran 𝑡𝑚 ω) ↔ 𝑈:ω⟶𝒫 ran 𝑡))
3625, 35mpbiri 248 . . 3 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → 𝑈 ∈ (𝒫 ran 𝑡𝑚 ω))
3712, 16, 36rspcdva 3455 . 2 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → (∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐) → ran 𝑈 ∈ ran 𝑈))
384, 37mpd 15 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 1632  wcel 2139  {cab 2746  wral 3050  Vcvv 3340  cin 3714  wss 3715  c0 4058  ifcif 4230  𝒫 cpw 4302   cuni 4588   cint 4627  ran crn 5267  suc csuc 5886   Fn wfn 6044  wf 6045  1-1wf1 6046  cfv 6049  (class class class)co 6814  cmpt2 6816  ωcom 7231  seq𝜔cseqom 7712  𝑚 cmap 8025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-seqom 7713  df-map 8027
This theorem is referenced by:  fin23lem21  9373
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