Theorem fin23lem17 10344

Description: Lemma for fin23 10395. 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 10338	. . 3 ⊢ (𝑐 ∈ ω → (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐))
3	2	rgen 3080	. 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 3967	. . . . 5 ⊢ (𝑏 = 𝑈 → ((𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) ↔ (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐)))
7	6	ralbidv 3187	. . . 4 ⊢ (𝑏 = 𝑈 → (∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) ↔ ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐)))
8		rneq 5924	. . . . . 6 ⊢ (𝑏 = 𝑈 → ran 𝑏 = ran 𝑈)
9	8	inteqd 4915	. . . . 5 ⊢ (𝑏 = 𝑈 → ∩ ran 𝑏 = ∩ ran 𝑈)
10	9, 8	eleq12d 2856	. . . 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 10335	. . . . 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 486	. . 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 8448	. . . . . 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 4909	. . . . . 6 ⊢ ran 𝑈 ⊆ 𝒫 ∪ ran 𝑈
20	1	fin23lem16 10341	. . . . . . 7 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡
21	20	pweqi 4576	. . . . . 6 ⊢ 𝒫 ∪ ran 𝑈 = 𝒫 ∪ ran 𝑡
22	19, 21	sseqtri 3982	. . . . 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 705	. . . 4 ⊢ 𝑈:ω⟶𝒫 ∪ ran 𝑡
25		vex 3457	. . . . . . . 8 ⊢ 𝑡 ∈ V
26	25	rnex 7911	. . . . . . 7 ⊢ ran 𝑡 ∈ V
27	26	uniex 7747	. . . . . 6 ⊢ ∪ ran 𝑡 ∈ V
28	27	pwex 5349	. . . . 5 ⊢ 𝒫 ∪ ran 𝑡 ∈ V
29		f1f 6775	. . . . . . 7 ⊢ (𝑡:ω–1-1→𝑉 → 𝑡:ω⟶𝑉)
30		dmfex 7906	. . . . . . 7 ⊢ ((𝑡 ∈ V ∧ 𝑡:ω⟶𝑉) → ω ∈ V)
31	25, 29, 30	sylancr 599	. . . . . 6 ⊢ (𝑡:ω–1-1→𝑉 → ω ∈ V)
32	31	adantl 487	. . . . 5 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ω ∈ V)
33		elmapg 8842	. . . . 5 ⊢ ((𝒫 ∪ ran 𝑡 ∈ V ∧ ω ∈ V) → (𝑈 ∈ (𝒫 ∪ ran 𝑡 ↑m ω) ↔ 𝑈:ω⟶𝒫 ∪ ran 𝑡))
34	28, 32, 33	sylancr 599	. . . 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 3580	. 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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740  ∀wral 3078  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  ifcif 4485  𝒫 cpw 4560  ∪ cuni 4870  ∩ cint 4910  ran crn 5660  suc csuc 6363   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  ωcom 7866  seq_ωcseqom 8440   ↑_m cmap 8830

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-seqom 8441  df-map 8832

This theorem is used by:  fin23lem21  10345