Theorem fin23lem21 9936

Description: Lemma for fin23 9986. 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 9935	. 2 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∩ ran 𝑈 ∈ ran 𝑈)
4	1	fnseqom 8180	. . . . 5 ⊢ 𝑈 Fn ω
5		fvelrnb 6762	. . . . 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 3405	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
9		f1f1orn 6661	. . . . . . . . . 10 ⊢ (𝑡:ω–1-1→𝑉 → 𝑡:ω–1-1-onto→ran 𝑡)
10		f1oen3g 8633	. . . . . . . . . 10 ⊢ ((𝑡 ∈ V ∧ 𝑡:ω–1-1-onto→ran 𝑡) → ω ≈ ran 𝑡)
11	8, 9, 10	sylancr 590	. . . . . . . . 9 ⊢ (𝑡:ω–1-1→𝑉 → ω ≈ ran 𝑡)
12		ominf 8879	. . . . . . . . 9 ⊢ ¬ ω ∈ Fin
13		ssdif0 4268	. . . . . . . . . . 11 ⊢ (ran 𝑡 ⊆ {∅} ↔ (ran 𝑡 ∖ {∅}) = ∅)
14		snfi 8710	. . . . . . . . . . . . 13 ⊢ {∅} ∈ Fin
15		ssfi 8840	. . . . . . . . . . . . 13 ⊢ (({∅} ∈ Fin ∧ ran 𝑡 ⊆ {∅}) → ran 𝑡 ∈ Fin)
16	14, 15	mpan 690	. . . . . . . . . . . 12 ⊢ (ran 𝑡 ⊆ {∅} → ran 𝑡 ∈ Fin)
17		enfi 8854	. . . . . . . . . . . 12 ⊢ (ω ≈ ran 𝑡 → (ω ∈ Fin ↔ ran 𝑡 ∈ Fin))
18	16, 17	syl5ibr 249	. . . . . . . . . . 11 ⊢ (ω ≈ ran 𝑡 → (ran 𝑡 ⊆ {∅} → ω ∈ Fin))
19	13, 18	syl5bir 246	. . . . . . . . . 10 ⊢ (ω ≈ ran 𝑡 → ((ran 𝑡 ∖ {∅}) = ∅ → ω ∈ Fin))
20	19	necon3bd 2949	. . . . . . . . 9 ⊢ (ω ≈ ran 𝑡 → (¬ ω ∈ Fin → (ran 𝑡 ∖ {∅}) ≠ ∅))
21	11, 12, 20	mpisyl 21	. . . . . . . 8 ⊢ (𝑡:ω–1-1→𝑉 → (ran 𝑡 ∖ {∅}) ≠ ∅)
22		n0 4251	. . . . . . . . 9 ⊢ ((ran 𝑡 ∖ {∅}) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}))
23		eldifsn 4690	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (ran 𝑡 ∖ {∅}) ↔ (𝑎 ∈ ran 𝑡 ∧ 𝑎 ≠ ∅))
24		elssuni 4841	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ran 𝑡 → 𝑎 ⊆ ∪ ran 𝑡)
25		ssn0 4305	. . . . . . . . . . . 12 ⊢ ((𝑎 ⊆ ∪ ran 𝑡 ∧ 𝑎 ≠ ∅) → ∪ ran 𝑡 ≠ ∅)
26	24, 25	sylan 583	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ran 𝑡 ∧ 𝑎 ≠ ∅) → ∪ ran 𝑡 ≠ ∅)
27	23, 26	sylbi 220	. . . . . . . . . 10 ⊢ (𝑎 ∈ (ran 𝑡 ∖ {∅}) → ∪ ran 𝑡 ≠ ∅)
28	27	exlimiv 1938	. . . . . . . . 9 ⊢ (∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}) → ∪ ran 𝑡 ≠ ∅)
29	22, 28	sylbi 220	. . . . . . . 8 ⊢ ((ran 𝑡 ∖ {∅}) ≠ ∅ → ∪ ran 𝑡 ≠ ∅)
30	21, 29	syl 17	. . . . . . 7 ⊢ (𝑡:ω–1-1→𝑉 → ∪ ran 𝑡 ≠ ∅)
31	1	fin23lem14 9930	. . . . . . 7 ⊢ ((𝑎 ∈ ω ∧ ∪ ran 𝑡 ≠ ∅) → (𝑈‘𝑎) ≠ ∅)
32	7, 30, 31	syl2anr 600	. . . . . 6 ⊢ ((𝑡:ω–1-1→𝑉 ∧ 𝑎 ∈ ω) → (𝑈‘𝑎) ≠ ∅)
33		neeq1 2997	. . . . . 6 ⊢ ((𝑈‘𝑎) = ∩ ran 𝑈 → ((𝑈‘𝑎) ≠ ∅ ↔ ∩ ran 𝑈 ≠ ∅))
34	32, 33	syl5ibcom 248	. . . . 5 ⊢ ((𝑡:ω–1-1→𝑉 ∧ 𝑎 ∈ ω) → ((𝑈‘𝑎) = ∩ ran 𝑈 → ∩ ran 𝑈 ≠ ∅))
35	34	rexlimdva 3196	. . . 4 ⊢ (𝑡:ω–1-1→𝑉 → (∃𝑎 ∈ ω (𝑈‘𝑎) = ∩ ran 𝑈 → ∩ ran 𝑈 ≠ ∅))
36	6, 35	syl5bi 245	. . 3 ⊢ (𝑡:ω–1-1→𝑉 → (∩ ran 𝑈 ∈ ran 𝑈 → ∩ ran 𝑈 ≠ ∅))
37	36	adantl 485	. 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 209   ∧ wa 399   = wceq 1543  ∃wex 1787   ∈ wcel 2110  {cab 2712   ≠ wne 2935  ∀wral 3054  ∃wrex 3055  Vcvv 3401   ∖ cdif 3854   ∩ cin 3856   ⊆ wss 3857  ∅c0 4227  ifcif 4429  𝒫 cpw 4503  {csn 4531  ∪ cuni 4809  ∩ cint 4849   class class class wbr 5043  ran crn 5541  suc csuc 6204   Fn wfn 6364  –1-1→wf1 6366  –1-1-onto→wf1o 6368  ‘cfv 6369  (class class class)co 7202   ∈ cmpo 7204  ωcom 7633  seq_ωcseqom 8172   ↑_m cmap 8497   ≈ cen 8612  Fincfn 8615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-seqom 8173  df-1o 8191  df-er 8380  df-map 8499  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619

This theorem is referenced by:  fin23lem31  9940