Theorem fin23lem21 9760

Description: Lemma for fin23 9810. 𝑋 is not empty. We only need here that 𝑡 has at least one set in its range besides ∅; the much stronger hypothesis here will serve as our induction hypothesis though. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 6-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
fin23lem21	⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∩ 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 fin23lem21

Step	Hyp	Ref	Expression
1		fin23lem.a	. . 3 ⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
2		fin23lem17.f	. . 3 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
3	1, 2	fin23lem17 9759	. 2 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∩ ran 𝑈 ∈ ran 𝑈)
4	1	fnseqom 8090	. . . . 5 ⊢ 𝑈 Fn ω
5		fvelrnb 6725	. . . . 5 ⊢ (𝑈 Fn ω → (∩ ran 𝑈 ∈ ran 𝑈 ↔ ∃𝑎 ∈ ω (𝑈‘𝑎) = ∩ ran 𝑈))
6	4, 5	ax-mp 5	. . . 4 ⊢ (∩ ran 𝑈 ∈ ran 𝑈 ↔ ∃𝑎 ∈ ω (𝑈‘𝑎) = ∩ ran 𝑈)
7		id 22	. . . . . . 7 ⊢ (𝑎 ∈ ω → 𝑎 ∈ ω)
8		vex 3497	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
9		f1f1orn 6625	. . . . . . . . . 10 ⊢ (𝑡:ω–1-1→𝑉 → 𝑡:ω–1-1-onto→ran 𝑡)
10		f1oen3g 8524	. . . . . . . . . 10 ⊢ ((𝑡 ∈ V ∧ 𝑡:ω–1-1-onto→ran 𝑡) → ω ≈ ran 𝑡)
11	8, 9, 10	sylancr 589	. . . . . . . . 9 ⊢ (𝑡:ω–1-1→𝑉 → ω ≈ ran 𝑡)
12		ominf 8729	. . . . . . . . 9 ⊢ ¬ ω ∈ Fin
13		ssdif0 4322	. . . . . . . . . . 11 ⊢ (ran 𝑡 ⊆ {∅} ↔ (ran 𝑡 ∖ {∅}) = ∅)
14		snfi 8593	. . . . . . . . . . . . 13 ⊢ {∅} ∈ Fin
15		ssfi 8737	. . . . . . . . . . . . 13 ⊢ (({∅} ∈ Fin ∧ ran 𝑡 ⊆ {∅}) → ran 𝑡 ∈ Fin)
16	14, 15	mpan 688	. . . . . . . . . . . 12 ⊢ (ran 𝑡 ⊆ {∅} → ran 𝑡 ∈ Fin)
17		enfi 8733	. . . . . . . . . . . 12 ⊢ (ω ≈ ran 𝑡 → (ω ∈ Fin ↔ ran 𝑡 ∈ Fin))
18	16, 17	syl5ibr 248	. . . . . . . . . . 11 ⊢ (ω ≈ ran 𝑡 → (ran 𝑡 ⊆ {∅} → ω ∈ Fin))
19	13, 18	syl5bir 245	. . . . . . . . . 10 ⊢ (ω ≈ ran 𝑡 → ((ran 𝑡 ∖ {∅}) = ∅ → ω ∈ Fin))
20	19	necon3bd 3030	. . . . . . . . 9 ⊢ (ω ≈ ran 𝑡 → (¬ ω ∈ Fin → (ran 𝑡 ∖ {∅}) ≠ ∅))
21	11, 12, 20	mpisyl 21	. . . . . . . 8 ⊢ (𝑡:ω–1-1→𝑉 → (ran 𝑡 ∖ {∅}) ≠ ∅)
22		n0 4309	. . . . . . . . 9 ⊢ ((ran 𝑡 ∖ {∅}) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}))
23		eldifsn 4718	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (ran 𝑡 ∖ {∅}) ↔ (𝑎 ∈ ran 𝑡 ∧ 𝑎 ≠ ∅))
24		elssuni 4867	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ran 𝑡 → 𝑎 ⊆ ∪ ran 𝑡)
25		ssn0 4353	. . . . . . . . . . . 12 ⊢ ((𝑎 ⊆ ∪ ran 𝑡 ∧ 𝑎 ≠ ∅) → ∪ ran 𝑡 ≠ ∅)
26	24, 25	sylan 582	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ran 𝑡 ∧ 𝑎 ≠ ∅) → ∪ ran 𝑡 ≠ ∅)
27	23, 26	sylbi 219	. . . . . . . . . 10 ⊢ (𝑎 ∈ (ran 𝑡 ∖ {∅}) → ∪ ran 𝑡 ≠ ∅)
28	27	exlimiv 1927	. . . . . . . . 9 ⊢ (∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}) → ∪ ran 𝑡 ≠ ∅)
29	22, 28	sylbi 219	. . . . . . . 8 ⊢ ((ran 𝑡 ∖ {∅}) ≠ ∅ → ∪ ran 𝑡 ≠ ∅)
30	21, 29	syl 17	. . . . . . 7 ⊢ (𝑡:ω–1-1→𝑉 → ∪ ran 𝑡 ≠ ∅)
31	1	fin23lem14 9754	. . . . . . 7 ⊢ ((𝑎 ∈ ω ∧ ∪ ran 𝑡 ≠ ∅) → (𝑈‘𝑎) ≠ ∅)
32	7, 30, 31	syl2anr 598	. . . . . 6 ⊢ ((𝑡:ω–1-1→𝑉 ∧ 𝑎 ∈ ω) → (𝑈‘𝑎) ≠ ∅)
33		neeq1 3078	. . . . . 6 ⊢ ((𝑈‘𝑎) = ∩ ran 𝑈 → ((𝑈‘𝑎) ≠ ∅ ↔ ∩ ran 𝑈 ≠ ∅))
34	32, 33	syl5ibcom 247	. . . . 5 ⊢ ((𝑡:ω–1-1→𝑉 ∧ 𝑎 ∈ ω) → ((𝑈‘𝑎) = ∩ ran 𝑈 → ∩ ran 𝑈 ≠ ∅))
35	34	rexlimdva 3284	. . . 4 ⊢ (𝑡:ω–1-1→𝑉 → (∃𝑎 ∈ ω (𝑈‘𝑎) = ∩ ran 𝑈 → ∩ ran 𝑈 ≠ ∅))
36	6, 35	syl5bi 244	. . 3 ⊢ (𝑡:ω–1-1→𝑉 → (∩ ran 𝑈 ∈ ran 𝑈 → ∩ ran 𝑈 ≠ ∅))
37	36	adantl 484	. 2 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → (∩ ran 𝑈 ∈ ran 𝑈 → ∩ ran 𝑈 ≠ ∅))
38	3, 37	mpd 15	1 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∩ ran 𝑈 ≠ ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  {cab 2799   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∖ cdif 3932   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290  ifcif 4466  𝒫 cpw 4538  {csn 4566  ∪ cuni 4837  ∩ cint 4875   class class class wbr 5065  ran crn 5555  suc csuc 6192   Fn wfn 6349  –1-1→wf1 6351  –1-1-onto→wf1o 6353  ‘cfv 6354  (class class class)co 7155   ∈ cmpo 7157  ωcom 7579  seq_ωcseqom 8082   ↑_m cmap 8405   ≈ cen 8505  Fincfn 8508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-seqom 8083  df-1o 8101  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512

This theorem is referenced by:  fin23lem31  9764