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Theorem fin23lem17 10347
Description: Lemma for fin23 10398. 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 10341 . . 3 (𝑐 ∈ ω → (𝑈‘suc 𝑐) ⊆ (𝑈𝑐))
32rgen 3058 . 2 𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)
4 fveq1 6890 . . . . . 6 (𝑏 = 𝑈 → (𝑏‘suc 𝑐) = (𝑈‘suc 𝑐))
5 fveq1 6890 . . . . . 6 (𝑏 = 𝑈 → (𝑏𝑐) = (𝑈𝑐))
64, 5sseq12d 4011 . . . . 5 (𝑏 = 𝑈 → ((𝑏‘suc 𝑐) ⊆ (𝑏𝑐) ↔ (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)))
76ralbidv 3172 . . . 4 (𝑏 = 𝑈 → (∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) ↔ ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)))
8 rneq 5932 . . . . . 6 (𝑏 = 𝑈 → ran 𝑏 = ran 𝑈)
98inteqd 4949 . . . . 5 (𝑏 = 𝑈 ran 𝑏 = ran 𝑈)
109, 8eleq12d 2822 . . . 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 10338 . . . . 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 8467 . . . . . 6 𝑈 Fn ω
17 dffn3 6729 . . . . . 6 (𝑈 Fn ω ↔ 𝑈:ω⟶ran 𝑈)
1816, 17mpbi 229 . . . . 5 𝑈:ω⟶ran 𝑈
19 pwuni 4943 . . . . . 6 ran 𝑈 ⊆ 𝒫 ran 𝑈
201fin23lem16 10344 . . . . . . 7 ran 𝑈 = ran 𝑡
2120pweqi 4614 . . . . . 6 𝒫 ran 𝑈 = 𝒫 ran 𝑡
2219, 21sseqtri 4014 . . . . 5 ran 𝑈 ⊆ 𝒫 ran 𝑡
23 fss 6733 . . . . 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 3473 . . . . . . . 8 𝑡 ∈ V
2625rnex 7910 . . . . . . 7 ran 𝑡 ∈ V
2726uniex 7738 . . . . . 6 ran 𝑡 ∈ V
2827pwex 5374 . . . . 5 𝒫 ran 𝑡 ∈ V
29 f1f 6787 . . . . . . 7 (𝑡:ω–1-1𝑉𝑡:ω⟶𝑉)
30 dmfex 7905 . . . . . . 7 ((𝑡 ∈ V ∧ 𝑡:ω⟶𝑉) → ω ∈ V)
3125, 29, 30sylancr 586 . . . . . 6 (𝑡:ω–1-1𝑉 → ω ∈ V)
3231adantl 481 . . . . 5 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ω ∈ V)
33 elmapg 8847 . . . . 5 ((𝒫 ran 𝑡 ∈ V ∧ ω ∈ V) → (𝑈 ∈ (𝒫 ran 𝑡m ω) ↔ 𝑈:ω⟶𝒫 ran 𝑡))
3428, 32, 33sylancr 586 . . . 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 3608 . 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 205  wa 395   = wceq 1534  wcel 2099  {cab 2704  wral 3056  Vcvv 3469  cin 3943  wss 3944  c0 4318  ifcif 4524  𝒫 cpw 4598   cuni 4903   cint 4944  ran crn 5673  suc csuc 6365   Fn wfn 6537  wf 6538  1-1wf1 6539  cfv 6542  (class class class)co 7414  cmpo 7416  ωcom 7862  seqωcseqom 8459  m cmap 8834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-seqom 8460  df-map 8836
This theorem is referenced by:  fin23lem21  10348
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