Theorem fin23lem17 10322

Description: Lemma for fin23 10373. 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 10316	. . 3 ⊢ (𝑐 ∈ ω → (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐))
3	2	rgen 3087	. 2 ⊢ ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐)
4		fveq1 6881	. . . . . 6 ⊢ (𝑏 = 𝑈 → (𝑏‘suc 𝑐) = (𝑈‘suc 𝑐))
5		fveq1 6881	. . . . . 6 ⊢ (𝑏 = 𝑈 → (𝑏‘𝑐) = (𝑈‘𝑐))
6	4, 5	sseq12d 3978	. . . . 5 ⊢ (𝑏 = 𝑈 → ((𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) ↔ (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐)))
7	6	ralbidv 3194	. . . 4 ⊢ (𝑏 = 𝑈 → (∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) ↔ ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐)))
8		rneq 5927	. . . . . 6 ⊢ (𝑏 = 𝑈 → ran 𝑏 = ran 𝑈)
9	8	inteqd 4921	. . . . 5 ⊢ (𝑏 = 𝑈 → ∩ ran 𝑏 = ∩ ran 𝑈)
10	9, 8	eleq12d 2863	. . . 4 ⊢ (𝑏 = 𝑈 → (∩ ran 𝑏 ∈ ran 𝑏 ↔ ∩ ran 𝑈 ∈ ran 𝑈))
11	7, 10	imbi12d 347	. . 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 10313	. . . . 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 270	. . . 4 ⊢ (∪ ran 𝑡 ∈ 𝐹 → ∀𝑏 ∈ (𝒫 ∪ ran 𝑡 ↑m ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) → ∩ ran 𝑏 ∈ ran 𝑏))
15	14	adantr 485	. . 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 8442	. . . . . 6 ⊢ 𝑈 Fn ω
17		dffn3 6719	. . . . . 6 ⊢ (𝑈 Fn ω ↔ 𝑈:ω⟶ran 𝑈)
18	16, 17	mpbi 233	. . . . 5 ⊢ 𝑈:ω⟶ran 𝑈
19		pwuni 4915	. . . . . 6 ⊢ ran 𝑈 ⊆ 𝒫 ∪ ran 𝑈
20	1	fin23lem16 10319	. . . . . . 7 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡
21	20	pweqi 4583	. . . . . 6 ⊢ 𝒫 ∪ ran 𝑈 = 𝒫 ∪ ran 𝑡
22	19, 21	sseqtri 3993	. . . . 5 ⊢ ran 𝑈 ⊆ 𝒫 ∪ ran 𝑡
23		fss 6723	. . . . 5 ⊢ ((𝑈:ω⟶ran 𝑈 ∧ ran 𝑈 ⊆ 𝒫 ∪ ran 𝑡) → 𝑈:ω⟶𝒫 ∪ ran 𝑡)
24	18, 22, 23	mp2an 704	. . . 4 ⊢ 𝑈:ω⟶𝒫 ∪ ran 𝑡
25		vex 3467	. . . . . . . 8 ⊢ 𝑡 ∈ V
26	25	rnex 7907	. . . . . . 7 ⊢ ran 𝑡 ∈ V
27	26	uniex 7740	. . . . . 6 ⊢ ∪ ran 𝑡 ∈ V
28	27	pwex 5352	. . . . 5 ⊢ 𝒫 ∪ ran 𝑡 ∈ V
29		f1f 6775	. . . . . . 7 ⊢ (𝑡:ω–1-1→𝑉 → 𝑡:ω⟶𝑉)
30		dmfex 7902	. . . . . . 7 ⊢ ((𝑡 ∈ V ∧ 𝑡:ω⟶𝑉) → ω ∈ V)
31	25, 29, 30	sylancr 598	. . . . . 6 ⊢ (𝑡:ω–1-1→𝑉 → ω ∈ V)
32	31	adantl 486	. . . . 5 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ω ∈ V)
33		elmapg 8836	. . . . 5 ⊢ ((𝒫 ∪ ran 𝑡 ∈ V ∧ ω ∈ V) → (𝑈 ∈ (𝒫 ∪ ran 𝑡 ↑m ω) ↔ 𝑈:ω⟶𝒫 ∪ ran 𝑡))
34	28, 32, 33	sylancr 598	. . . 4 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → (𝑈 ∈ (𝒫 ∪ ran 𝑡 ↑m ω) ↔ 𝑈:ω⟶𝒫 ∪ ran 𝑡))
35	24, 34	mpbiri 261	. . 3 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → 𝑈 ∈ (𝒫 ∪ ran 𝑡 ↑m ω))
36	11, 15, 35	rspcdva 3591	. 2 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → (∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐) → ∩ ran 𝑈 ∈ ran 𝑈))
37	3, 36	mpi 21	1 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∩ ran 𝑈 ∈ ran 𝑈)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  ifcif 4492  𝒫 cpw 4567  ∪ cuni 4876  ∩ cint 4916  ran crn 5663  suc csuc 6363   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  ωcom 7862  seq_ωcseqom 8434   ↑_m cmap 8824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-seqom 8435  df-map 8826

This theorem is referenced by:  fin23lem21  10323