Theorem fin23lem17 9043

Description: Lemma for fin23 9094. 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	. . . . . 6 ⊢ 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
2	1	fnseqom 7437	. . . . 5 ⊢ 𝑈 Fn ω
3		dffn3 5967	. . . . 5 ⊢ (𝑈 Fn ω ↔ 𝑈:ω⟶ran 𝑈)
4	2, 3	mpbi 219	. . . 4 ⊢ 𝑈:ω⟶ran 𝑈
5		pwuni 4825	. . . . 5 ⊢ ran 𝑈 ⊆ 𝒫 ∪ ran 𝑈
6	1	fin23lem16 9040	. . . . . 6 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡
7	6	pweqi 4112	. . . . 5 ⊢ 𝒫 ∪ ran 𝑈 = 𝒫 ∪ ran 𝑡
8	5, 7	sseqtri 3600	. . . 4 ⊢ ran 𝑈 ⊆ 𝒫 ∪ ran 𝑡
9		fss 5969	. . . 4 ⊢ ((𝑈:ω⟶ran 𝑈 ∧ ran 𝑈 ⊆ 𝒫 ∪ ran 𝑡) → 𝑈:ω⟶𝒫 ∪ ran 𝑡)
10	4, 8, 9	mp2an 704	. . 3 ⊢ 𝑈:ω⟶𝒫 ∪ ran 𝑡
11		vex 3176	. . . . . . 7 ⊢ 𝑡 ∈ V
12	11	rnex 6992	. . . . . 6 ⊢ ran 𝑡 ∈ V
13	12	uniex 6851	. . . . 5 ⊢ ∪ ran 𝑡 ∈ V
14	13	pwex 4774	. . . 4 ⊢ 𝒫 ∪ ran 𝑡 ∈ V
15		f1f 6014	. . . . . 6 ⊢ (𝑡:ω–1-1→𝑉 → 𝑡:ω⟶𝑉)
16		dmfex 7017	. . . . . 6 ⊢ ((𝑡 ∈ V ∧ 𝑡:ω⟶𝑉) → ω ∈ V)
17	11, 15, 16	sylancr 694	. . . . 5 ⊢ (𝑡:ω–1-1→𝑉 → ω ∈ V)
18	17	adantl 481	. . . 4 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ω ∈ V)
19		elmapg 7757	. . . 4 ⊢ ((𝒫 ∪ ran 𝑡 ∈ V ∧ ω ∈ V) → (𝑈 ∈ (𝒫 ∪ ran 𝑡 ↑𝑚 ω) ↔ 𝑈:ω⟶𝒫 ∪ ran 𝑡))
20	14, 18, 19	sylancr 694	. . 3 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → (𝑈 ∈ (𝒫 ∪ ran 𝑡 ↑𝑚 ω) ↔ 𝑈:ω⟶𝒫 ∪ ran 𝑡))
21	10, 20	mpbiri 247	. 2 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → 𝑈 ∈ (𝒫 ∪ ran 𝑡 ↑𝑚 ω))
22		fin23lem17.f	. . . . 5 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
23	22	isfin3ds 9034	. . . 4 ⊢ (∪ ran 𝑡 ∈ 𝐹 → (∪ ran 𝑡 ∈ 𝐹 ↔ ∀𝑏 ∈ (𝒫 ∪ ran 𝑡 ↑𝑚 ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) → ∩ ran 𝑏 ∈ ran 𝑏)))
24	23	ibi 255	. . 3 ⊢ (∪ ran 𝑡 ∈ 𝐹 → ∀𝑏 ∈ (𝒫 ∪ ran 𝑡 ↑𝑚 ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) → ∩ ran 𝑏 ∈ ran 𝑏))
25	24	adantr 480	. 2 ⊢ ((∪ 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))
26	1	fin23lem13 9037	. . . 4 ⊢ (𝑐 ∈ ω → (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐))
27	26	rgen 2906	. . 3 ⊢ ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐)
28	27	a1i 11	. 2 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐))
29		fveq1 6102	. . . . . 6 ⊢ (𝑏 = 𝑈 → (𝑏‘suc 𝑐) = (𝑈‘suc 𝑐))
30		fveq1 6102	. . . . . 6 ⊢ (𝑏 = 𝑈 → (𝑏‘𝑐) = (𝑈‘𝑐))
31	29, 30	sseq12d 3597	. . . . 5 ⊢ (𝑏 = 𝑈 → ((𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) ↔ (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐)))
32	31	ralbidv 2969	. . . 4 ⊢ (𝑏 = 𝑈 → (∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) ↔ ∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐)))
33		rneq 5272	. . . . . 6 ⊢ (𝑏 = 𝑈 → ran 𝑏 = ran 𝑈)
34	33	inteqd 4415	. . . . 5 ⊢ (𝑏 = 𝑈 → ∩ ran 𝑏 = ∩ ran 𝑈)
35	34, 33	eleq12d 2682	. . . 4 ⊢ (𝑏 = 𝑈 → (∩ ran 𝑏 ∈ ran 𝑏 ↔ ∩ ran 𝑈 ∈ ran 𝑈))
36	32, 35	imbi12d 333	. . 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)))
37	36	rspcv 3278	. 2 ⊢ (𝑈 ∈ (𝒫 ∪ ran 𝑡 ↑𝑚 ω) → (∀𝑏 ∈ (𝒫 ∪ ran 𝑡 ↑𝑚 ω)(∀𝑐 ∈ ω (𝑏‘suc 𝑐) ⊆ (𝑏‘𝑐) → ∩ ran 𝑏 ∈ ran 𝑏) → (∀𝑐 ∈ ω (𝑈‘suc 𝑐) ⊆ (𝑈‘𝑐) → ∩ ran 𝑈 ∈ ran 𝑈)))
38	21, 25, 28, 37	syl3c 64	1 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∩ ran 𝑈 ∈ ran 𝑈)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  ∪ cuni 4372  ∩ cint 4410  ran crn 5039  suc csuc 5642   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957  seq_𝜔cseqom 7429   ↑_𝑚 cmap 7744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-map 7746

This theorem is referenced by:  fin23lem21  9044