Theorem fin23lem21 10360

Description: Lemma for fin23 10410. 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 10359	. 2 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∩ ran 𝑈 ∈ ran 𝑈)
4	1	fnseqom 8472	. . . . 5 ⊢ 𝑈 Fn ω
5		fvelrnb 6953	. . . . 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 3467	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
9		f1f1orn 6844	. . . . . . . . . 10 ⊢ (𝑡:ω–1-1→𝑉 → 𝑡:ω–1-1-onto→ran 𝑡)
10		f1oen3g 8983	. . . . . . . . . 10 ⊢ ((𝑡 ∈ V ∧ 𝑡:ω–1-1-onto→ran 𝑡) → ω ≈ ran 𝑡)
11	8, 9, 10	sylancr 585	. . . . . . . . 9 ⊢ (𝑡:ω–1-1→𝑉 → ω ≈ ran 𝑡)
12		ominf 9279	. . . . . . . . 9 ⊢ ¬ ω ∈ Fin
13		ssdif0 4359	. . . . . . . . . . 11 ⊢ (ran 𝑡 ⊆ {∅} ↔ (ran 𝑡 ∖ {∅}) = ∅)
14		snfi 9065	. . . . . . . . . . . . 13 ⊢ {∅} ∈ Fin
15		ssfi 9194	. . . . . . . . . . . . 13 ⊢ (({∅} ∈ Fin ∧ ran 𝑡 ⊆ {∅}) → ran 𝑡 ∈ Fin)
16	14, 15	mpan 688	. . . . . . . . . . . 12 ⊢ (ran 𝑡 ⊆ {∅} → ran 𝑡 ∈ Fin)
17		enfi 9211	. . . . . . . . . . . 12 ⊢ (ω ≈ ran 𝑡 → (ω ∈ Fin ↔ ran 𝑡 ∈ Fin))
18	16, 17	imbitrrid 245	. . . . . . . . . . 11 ⊢ (ω ≈ ran 𝑡 → (ran 𝑡 ⊆ {∅} → ω ∈ Fin))
19	13, 18	biimtrrid 242	. . . . . . . . . 10 ⊢ (ω ≈ ran 𝑡 → ((ran 𝑡 ∖ {∅}) = ∅ → ω ∈ Fin))
20	19	necon3bd 2944	. . . . . . . . 9 ⊢ (ω ≈ ran 𝑡 → (¬ ω ∈ Fin → (ran 𝑡 ∖ {∅}) ≠ ∅))
21	11, 12, 20	mpisyl 21	. . . . . . . 8 ⊢ (𝑡:ω–1-1→𝑉 → (ran 𝑡 ∖ {∅}) ≠ ∅)
22		n0 4342	. . . . . . . . 9 ⊢ ((ran 𝑡 ∖ {∅}) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}))
23		eldifsn 4786	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (ran 𝑡 ∖ {∅}) ↔ (𝑎 ∈ ran 𝑡 ∧ 𝑎 ≠ ∅))
24		elssuni 4935	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ran 𝑡 → 𝑎 ⊆ ∪ ran 𝑡)
25		ssn0 4396	. . . . . . . . . . . 12 ⊢ ((𝑎 ⊆ ∪ ran 𝑡 ∧ 𝑎 ≠ ∅) → ∪ ran 𝑡 ≠ ∅)
26	24, 25	sylan 578	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ran 𝑡 ∧ 𝑎 ≠ ∅) → ∪ ran 𝑡 ≠ ∅)
27	23, 26	sylbi 216	. . . . . . . . . 10 ⊢ (𝑎 ∈ (ran 𝑡 ∖ {∅}) → ∪ ran 𝑡 ≠ ∅)
28	27	exlimiv 1925	. . . . . . . . 9 ⊢ (∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}) → ∪ ran 𝑡 ≠ ∅)
29	22, 28	sylbi 216	. . . . . . . 8 ⊢ ((ran 𝑡 ∖ {∅}) ≠ ∅ → ∪ ran 𝑡 ≠ ∅)
30	21, 29	syl 17	. . . . . . 7 ⊢ (𝑡:ω–1-1→𝑉 → ∪ ran 𝑡 ≠ ∅)
31	1	fin23lem14 10354	. . . . . . 7 ⊢ ((𝑎 ∈ ω ∧ ∪ ran 𝑡 ≠ ∅) → (𝑈‘𝑎) ≠ ∅)
32	7, 30, 31	syl2anr 595	. . . . . 6 ⊢ ((𝑡:ω–1-1→𝑉 ∧ 𝑎 ∈ ω) → (𝑈‘𝑎) ≠ ∅)
33		neeq1 2993	. . . . . 6 ⊢ ((𝑈‘𝑎) = ∩ ran 𝑈 → ((𝑈‘𝑎) ≠ ∅ ↔ ∩ ran 𝑈 ≠ ∅))
34	32, 33	syl5ibcom 244	. . . . 5 ⊢ ((𝑡:ω–1-1→𝑉 ∧ 𝑎 ∈ ω) → ((𝑈‘𝑎) = ∩ ran 𝑈 → ∩ ran 𝑈 ≠ ∅))
35	34	rexlimdva 3145	. . . 4 ⊢ (𝑡:ω–1-1→𝑉 → (∃𝑎 ∈ ω (𝑈‘𝑎) = ∩ ran 𝑈 → ∩ ran 𝑈 ≠ ∅))
36	6, 35	biimtrid 241	. . 3 ⊢ (𝑡:ω–1-1→𝑉 → (∩ ran 𝑈 ∈ ran 𝑈 → ∩ ran 𝑈 ≠ ∅))
37	36	adantl 480	. 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 205   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2702   ≠ wne 2930  ∀wral 3051  ∃wrex 3060  Vcvv 3463   ∖ cdif 3937   ∩ cin 3939   ⊆ wss 3940  ∅c0 4318  ifcif 4524  𝒫 cpw 4598  {csn 4624  ∪ cuni 4903  ∩ cint 4944   class class class wbr 5143  ran crn 5673  suc csuc 6366   Fn wfn 6537  –1-1→wf1 6539  –1-1-onto→wf1o 6541  ‘cfv 6542  (class class class)co 7415   ∈ cmpo 7417  ωcom 7867  seq_ωcseqom 8464   ↑_m cmap 8841   ≈ cen 8957  Fincfn 8960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-seqom 8465  df-1o 8483  df-map 8843  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964

This theorem is referenced by:  fin23lem31  10364