Theorem fin23lem17 10251

Description: Lemma for fin23 10302. 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(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
fin23lem17.f	⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}

Assertion

Ref	Expression
fin23lem17	⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∩ 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.

Step	Hyp	Ref	Expression
1		fin23lem.a	. . . 4 ⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
2	1	fin23lem13 10245	. . 3 ⊢ (𝑐 ∈ ω → (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐))
3	2	rgen 3055	. 2 ⊢ ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐)
4		fveq1 6826	. . . . . 6 ⊢ (𝑏 = 𝑈 → (𝑏‘suc 𝑐) = (𝑈‘suc 𝑐))
5		fveq1 6826	. . . . . 6 ⊢ (𝑏 = 𝑈 → (𝑏‘𝑐) = (𝑈‘𝑐))
6	4, 5	sseq12d 3948	. . . . 5 ⊢ (𝑏 = 𝑈 → ((𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) ↔ (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐)))
7	6	ralbidv 3162	. . . 4 ⊢ (𝑏 = 𝑈 → (∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) ↔ ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐)))
8		rneq 5878	. . . . . 6 ⊢ (𝑏 = 𝑈 → ran 𝑏 = ran 𝑈)
9	8	inteqd 4882	. . . . 5 ⊢ (𝑏 = 𝑈 → ∩ ran 𝑏 = ∩ ran 𝑈)
10	9, 8	eleq12d 2833	. . . 4 ⊢ (𝑏 = 𝑈 → (∩ ran 𝑏 ∈ ran 𝑏 ↔ ∩ ran 𝑈 ∈ ran 𝑈))
11	7, 10	imbi12d 345	. . 3 ⊢ (𝑏 = 𝑈 → ((∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) → ∩ ran 𝑏 ∈ ran 𝑏) ↔ (∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐) → ∩ ran 𝑈 ∈ ran 𝑈)))
12		fin23lem17.f	. . . . . 6 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
13	12	isfin3ds 10242	. . . . 5 ⊢ (∪ ran 𝑡 ∈ 𝐹 → (∪ ran 𝑡 ∈ 𝐹 ↔ ∀𝑏 ∈ (𝒫 ∪ ran 𝑡 ↑m ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) → ∩ ran 𝑏 ∈ ran 𝑏)))
14	13	ibi 268	. . . 4 ⊢ (∪ ran 𝑡 ∈ 𝐹 → ∀𝑏 ∈ (𝒫 ∪ ran 𝑡 ↑m ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) → ∩ ran 𝑏 ∈ ran 𝑏))
15	14	adantr 481	. . 3 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∀𝑏 ∈ (𝒫 ∪ ran 𝑡 ↑m ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) → ∩ ran 𝑏 ∈ ran 𝑏))
16	1	fnseqom 8384	. . . . . 6 ⊢ 𝑈 Fn ω
17		dffn3 6667	. . . . . 6 ⊢ (𝑈 Fn ω ↔ 𝑈:ω⟶ran 𝑈)
18	16, 17	mpbi 231	. . . . 5 ⊢ 𝑈:ω⟶ran 𝑈
19		pwuni 4876	. . . . . 6 ⊢ ran 𝑈 ⊆ 𝒫 ∪ ran 𝑈
20	1	fin23lem16 10248	. . . . . . 7 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡
21	20	pweqi 4545	. . . . . 6 ⊢ 𝒫 ∪ ran 𝑈 = 𝒫 ∪ ran 𝑡
22	19, 21	sseqtri 3963	. . . . 5 ⊢ ran 𝑈 ⊆ 𝒫 ∪ ran 𝑡
23		fss 6671	. . . . 5 ⊢ ((𝑈:ω⟶ran 𝑈 ∧ ran 𝑈 ⊆ 𝒫 ∪ ran 𝑡) → 𝑈:ω⟶𝒫 ∪ ran 𝑡)
24	18, 22, 23	mp2an 698	. . . 4 ⊢ 𝑈:ω⟶𝒫 ∪ ran 𝑡
25		vex 3435	. . . . . . . 8 ⊢ 𝑡 ∈ V
26	25	rnex 7850	. . . . . . 7 ⊢ ran 𝑡 ∈ V
27	26	uniex 7684	. . . . . 6 ⊢ ∪ ran 𝑡 ∈ V
28	27	pwex 5309	. . . . 5 ⊢ 𝒫 ∪ ran 𝑡 ∈ V
29		f1f 6723	. . . . . . 7 ⊢ (𝑡:ω–1-1→𝑉 → 𝑡:ω⟶𝑉)
30		dmfex 7845	. . . . . . 7 ⊢ ((𝑡 ∈ V ∧ 𝑡:ω⟶𝑉) → ω ∈ V)
31	25, 29, 30	sylancr 593	. . . . . 6 ⊢ (𝑡:ω–1-1→𝑉 → ω ∈ V)
32	31	adantl 482	. . . . 5 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ω ∈ V)
33		elmapg 8776	. . . . 5 ⊢ ((𝒫 ∪ ran 𝑡 ∈ V ∧ ω ∈ V) → (𝑈 ∈ (𝒫 ∪ ran 𝑡 ↑m ω) ↔ 𝑈:ω⟶𝒫 ∪ ran 𝑡))
34	28, 32, 33	sylancr 593	. . . 4 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → (𝑈 ∈ (𝒫 ∪ ran 𝑡 ↑m ω) ↔ 𝑈:ω⟶𝒫 ∪ ran 𝑡))
35	24, 34	mpbiri 259	. . 3 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → 𝑈 ∈ (𝒫 ∪ ran 𝑡 ↑m ω))
36	11, 15, 35	rspcdva 3561	. 2 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → (∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐) → ∩ ran 𝑈 ∈ ran 𝑈))
37	3, 36	mpi 20	1 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∩ ran 𝑈 ∈ ran 𝑈)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {cab 2717  ∀wral 3053  Vcvv 3431   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  ifcif 4454  𝒫 cpw 4529  ∪ cuni 4838  ∩ cint 4877  ran crn 5619  suc csuc 6312   Fn wfn 6480  ⟶wf 6481  –1-1→wf1 6482  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  ωcom 7806  seq_ωcseqom 8376   ↑_m cmap 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-seqom 8377  df-map 8765

This theorem is referenced by:  fin23lem21  10252