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Theorem fin23lem17 9105
Description: Lemma for fin23 9156. 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 7496 . . . . 5 𝑈 Fn ω
3 dffn3 6013 . . . . 5 (𝑈 Fn ω ↔ 𝑈:ω⟶ran 𝑈)
42, 3mpbi 220 . . . 4 𝑈:ω⟶ran 𝑈
5 pwuni 4864 . . . . 5 ran 𝑈 ⊆ 𝒫 ran 𝑈
61fin23lem16 9102 . . . . . 6 ran 𝑈 = ran 𝑡
76pweqi 4139 . . . . 5 𝒫 ran 𝑈 = 𝒫 ran 𝑡
85, 7sseqtri 3621 . . . 4 ran 𝑈 ⊆ 𝒫 ran 𝑡
9 fss 6015 . . . 4 ((𝑈:ω⟶ran 𝑈 ∧ ran 𝑈 ⊆ 𝒫 ran 𝑡) → 𝑈:ω⟶𝒫 ran 𝑡)
104, 8, 9mp2an 707 . . 3 𝑈:ω⟶𝒫 ran 𝑡
11 vex 3194 . . . . . . 7 𝑡 ∈ V
1211rnex 7048 . . . . . 6 ran 𝑡 ∈ V
1312uniex 6907 . . . . 5 ran 𝑡 ∈ V
1413pwex 4813 . . . 4 𝒫 ran 𝑡 ∈ V
15 f1f 6060 . . . . . 6 (𝑡:ω–1-1𝑉𝑡:ω⟶𝑉)
16 dmfex 7074 . . . . . 6 ((𝑡 ∈ V ∧ 𝑡:ω⟶𝑉) → ω ∈ V)
1711, 15, 16sylancr 694 . . . . 5 (𝑡:ω–1-1𝑉 → ω ∈ V)
1817adantl 482 . . . 4 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ω ∈ V)
19 elmapg 7816 . . . 4 ((𝒫 ran 𝑡 ∈ V ∧ ω ∈ V) → (𝑈 ∈ (𝒫 ran 𝑡𝑚 ω) ↔ 𝑈:ω⟶𝒫 ran 𝑡))
2014, 18, 19sylancr 694 . . 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 9096 . . . 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 481 . 2 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ∀𝑏 ∈ (𝒫 ran 𝑡𝑚 ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏))
261fin23lem13 9099 . . . 4 (𝑐 ∈ ω → (𝑈‘suc 𝑐) ⊆ (𝑈𝑐))
2726rgen 2922 . . 3 𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)
2827a1i 11 . 2 (( ran 𝑡𝐹𝑡:ω–1-1𝑉) → ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐))
29 fveq1 6149 . . . . . 6 (𝑏 = 𝑈 → (𝑏‘suc 𝑐) = (𝑈‘suc 𝑐))
30 fveq1 6149 . . . . . 6 (𝑏 = 𝑈 → (𝑏𝑐) = (𝑈𝑐))
3129, 30sseq12d 3618 . . . . 5 (𝑏 = 𝑈 → ((𝑏‘suc 𝑐) ⊆ (𝑏𝑐) ↔ (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)))
3231ralbidv 2985 . . . 4 (𝑏 = 𝑈 → (∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) ↔ ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐)))
33 rneq 5315 . . . . . 6 (𝑏 = 𝑈 → ran 𝑏 = ran 𝑈)
3433inteqd 4450 . . . . 5 (𝑏 = 𝑈 ran 𝑏 = ran 𝑈)
3534, 33eleq12d 2698 . . . 4 (𝑏 = 𝑈 → ( ran 𝑏 ∈ ran 𝑏 ran 𝑈 ∈ ran 𝑈))
3632, 35imbi12d 334 . . 3 (𝑏 = 𝑈 → ((∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏𝑐) → ran 𝑏 ∈ ran 𝑏) ↔ (∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈𝑐) → ran 𝑈 ∈ ran 𝑈)))
3736rspcv 3296 . 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 384   = wceq 1480  wcel 1992  {cab 2612  wral 2912  Vcvv 3191  cin 3559  wss 3560  c0 3896  ifcif 4063  𝒫 cpw 4135   cuni 4407   cint 4445  ran crn 5080  suc csuc 5687   Fn wfn 5845  wf 5846  1-1wf1 5847  cfv 5850  (class class class)co 6605  cmpt2 6607  ωcom 7013  seq𝜔cseqom 7488  𝑚 cmap 7803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-seqom 7489  df-map 7805
This theorem is referenced by:  fin23lem21  9106
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