Theorem fin23lem17 10333

Description: Lemma for fin23 10384. 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 10327	. . 3 ⊢ (𝑐 ∈ ω → (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐))
3	2	rgen 3064	. 2 ⊢ ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐)
4		fveq1 6891	. . . . . 6 ⊢ (𝑏 = 𝑈 → (𝑏‘suc 𝑐) = (𝑈‘suc 𝑐))
5		fveq1 6891	. . . . . 6 ⊢ (𝑏 = 𝑈 → (𝑏‘𝑐) = (𝑈‘𝑐))
6	4, 5	sseq12d 4016	. . . . 5 ⊢ (𝑏 = 𝑈 → ((𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) ↔ (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐)))
7	6	ralbidv 3178	. . . 4 ⊢ (𝑏 = 𝑈 → (∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) ↔ ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐)))
8		rneq 5936	. . . . . 6 ⊢ (𝑏 = 𝑈 → ran 𝑏 = ran 𝑈)
9	8	inteqd 4956	. . . . 5 ⊢ (𝑏 = 𝑈 → ∩ ran 𝑏 = ∩ ran 𝑈)
10	9, 8	eleq12d 2828	. . . 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 10324	. . . . 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 267	. . . 4 ⊢ (∪ ran 𝑡 ∈ 𝐹 → ∀𝑏 ∈ (𝒫 ∪ ran 𝑡 ↑m ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) → ∩ ran 𝑏 ∈ ran 𝑏))
15	14	adantr 482	. . 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 8455	. . . . . 6 ⊢ 𝑈 Fn ω
17		dffn3 6731	. . . . . 6 ⊢ (𝑈 Fn ω ↔ 𝑈:ω⟶ran 𝑈)
18	16, 17	mpbi 229	. . . . 5 ⊢ 𝑈:ω⟶ran 𝑈
19		pwuni 4950	. . . . . 6 ⊢ ran 𝑈 ⊆ 𝒫 ∪ ran 𝑈
20	1	fin23lem16 10330	. . . . . . 7 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡
21	20	pweqi 4619	. . . . . 6 ⊢ 𝒫 ∪ ran 𝑈 = 𝒫 ∪ ran 𝑡
22	19, 21	sseqtri 4019	. . . . 5 ⊢ ran 𝑈 ⊆ 𝒫 ∪ ran 𝑡
23		fss 6735	. . . . 5 ⊢ ((𝑈:ω⟶ran 𝑈 ∧ ran 𝑈 ⊆ 𝒫 ∪ ran 𝑡) → 𝑈:ω⟶𝒫 ∪ ran 𝑡)
24	18, 22, 23	mp2an 691	. . . 4 ⊢ 𝑈:ω⟶𝒫 ∪ ran 𝑡
25		vex 3479	. . . . . . . 8 ⊢ 𝑡 ∈ V
26	25	rnex 7903	. . . . . . 7 ⊢ ran 𝑡 ∈ V
27	26	uniex 7731	. . . . . 6 ⊢ ∪ ran 𝑡 ∈ V
28	27	pwex 5379	. . . . 5 ⊢ 𝒫 ∪ ran 𝑡 ∈ V
29		f1f 6788	. . . . . . 7 ⊢ (𝑡:ω–1-1→𝑉 → 𝑡:ω⟶𝑉)
30		dmfex 7898	. . . . . . 7 ⊢ ((𝑡 ∈ V ∧ 𝑡:ω⟶𝑉) → ω ∈ V)
31	25, 29, 30	sylancr 588	. . . . . 6 ⊢ (𝑡:ω–1-1→𝑉 → ω ∈ V)
32	31	adantl 483	. . . . 5 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ω ∈ V)
33		elmapg 8833	. . . . 5 ⊢ ((𝒫 ∪ ran 𝑡 ∈ V ∧ ω ∈ V) → (𝑈 ∈ (𝒫 ∪ ran 𝑡 ↑m ω) ↔ 𝑈:ω⟶𝒫 ∪ ran 𝑡))
34	28, 32, 33	sylancr 588	. . . 4 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → (𝑈 ∈ (𝒫 ∪ ran 𝑡 ↑m ω) ↔ 𝑈:ω⟶𝒫 ∪ ran 𝑡))
35	24, 34	mpbiri 258	. . 3 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → 𝑈 ∈ (𝒫 ∪ ran 𝑡 ↑m ω))
36	11, 15, 35	rspcdva 3614	. 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 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  ∀wral 3062  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949  ∅c0 4323  ifcif 4529  𝒫 cpw 4603  ∪ cuni 4909  ∩ cint 4951  ran crn 5678  suc csuc 6367   Fn wfn 6539  ⟶wf 6540  –1-1→wf1 6541  ‘cfv 6544  (class class class)co 7409   ∈ cmpo 7411  ωcom 7855  seq_ωcseqom 8447   ↑_m cmap 8820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-seqom 8448  df-map 8822

This theorem is referenced by:  fin23lem21  10334