Theorem fin23lem21 9044

Description: Lemma for fin23 9094. 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 9043	. 2 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∩ ran 𝑈 ∈ ran 𝑈)
4	1	fnseqom 7437	. . . . 5 ⊢ 𝑈 Fn ω
5		fvelrnb 6153	. . . . 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 3176	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
9		f1f1orn 6061	. . . . . . . . . 10 ⊢ (𝑡:ω–1-1→𝑉 → 𝑡:ω–1-1-onto→ran 𝑡)
10		f1oen3g 7857	. . . . . . . . . 10 ⊢ ((𝑡 ∈ V ∧ 𝑡:ω–1-1-onto→ran 𝑡) → ω ≈ ran 𝑡)
11	8, 9, 10	sylancr 694	. . . . . . . . 9 ⊢ (𝑡:ω–1-1→𝑉 → ω ≈ ran 𝑡)
12		ominf 8057	. . . . . . . . 9 ⊢ ¬ ω ∈ Fin
13		ssdif0 3896	. . . . . . . . . . 11 ⊢ (ran 𝑡 ⊆ {∅} ↔ (ran 𝑡 ∖ {∅}) = ∅)
14		snfi 7923	. . . . . . . . . . . . 13 ⊢ {∅} ∈ Fin
15		ssfi 8065	. . . . . . . . . . . . 13 ⊢ (({∅} ∈ Fin ∧ ran 𝑡 ⊆ {∅}) → ran 𝑡 ∈ Fin)
16	14, 15	mpan 702	. . . . . . . . . . . 12 ⊢ (ran 𝑡 ⊆ {∅} → ran 𝑡 ∈ Fin)
17		enfi 8061	. . . . . . . . . . . 12 ⊢ (ω ≈ ran 𝑡 → (ω ∈ Fin ↔ ran 𝑡 ∈ Fin))
18	16, 17	syl5ibr 235	. . . . . . . . . . 11 ⊢ (ω ≈ ran 𝑡 → (ran 𝑡 ⊆ {∅} → ω ∈ Fin))
19	13, 18	syl5bir 232	. . . . . . . . . 10 ⊢ (ω ≈ ran 𝑡 → ((ran 𝑡 ∖ {∅}) = ∅ → ω ∈ Fin))
20	19	necon3bd 2796	. . . . . . . . 9 ⊢ (ω ≈ ran 𝑡 → (¬ ω ∈ Fin → (ran 𝑡 ∖ {∅}) ≠ ∅))
21	11, 12, 20	mpisyl 21	. . . . . . . 8 ⊢ (𝑡:ω–1-1→𝑉 → (ran 𝑡 ∖ {∅}) ≠ ∅)
22		n0 3890	. . . . . . . . 9 ⊢ ((ran 𝑡 ∖ {∅}) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}))
23		eldifsn 4260	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (ran 𝑡 ∖ {∅}) ↔ (𝑎 ∈ ran 𝑡 ∧ 𝑎 ≠ ∅))
24		elssuni 4403	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ran 𝑡 → 𝑎 ⊆ ∪ ran 𝑡)
25		ssn0 3928	. . . . . . . . . . . 12 ⊢ ((𝑎 ⊆ ∪ ran 𝑡 ∧ 𝑎 ≠ ∅) → ∪ ran 𝑡 ≠ ∅)
26	24, 25	sylan 487	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ran 𝑡 ∧ 𝑎 ≠ ∅) → ∪ ran 𝑡 ≠ ∅)
27	23, 26	sylbi 206	. . . . . . . . . 10 ⊢ (𝑎 ∈ (ran 𝑡 ∖ {∅}) → ∪ ran 𝑡 ≠ ∅)
28	27	exlimiv 1845	. . . . . . . . 9 ⊢ (∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}) → ∪ ran 𝑡 ≠ ∅)
29	22, 28	sylbi 206	. . . . . . . 8 ⊢ ((ran 𝑡 ∖ {∅}) ≠ ∅ → ∪ ran 𝑡 ≠ ∅)
30	21, 29	syl 17	. . . . . . 7 ⊢ (𝑡:ω–1-1→𝑉 → ∪ ran 𝑡 ≠ ∅)
31	1	fin23lem14 9038	. . . . . . 7 ⊢ ((𝑎 ∈ ω ∧ ∪ ran 𝑡 ≠ ∅) → (𝑈‘𝑎) ≠ ∅)
32	7, 30, 31	syl2anr 494	. . . . . 6 ⊢ ((𝑡:ω–1-1→𝑉 ∧ 𝑎 ∈ ω) → (𝑈‘𝑎) ≠ ∅)
33		neeq1 2844	. . . . . 6 ⊢ ((𝑈‘𝑎) = ∩ ran 𝑈 → ((𝑈‘𝑎) ≠ ∅ ↔ ∩ ran 𝑈 ≠ ∅))
34	32, 33	syl5ibcom 234	. . . . 5 ⊢ ((𝑡:ω–1-1→𝑉 ∧ 𝑎 ∈ ω) → ((𝑈‘𝑎) = ∩ ran 𝑈 → ∩ ran 𝑈 ≠ ∅))
35	34	rexlimdva 3013	. . . 4 ⊢ (𝑡:ω–1-1→𝑉 → (∃𝑎 ∈ ω (𝑈‘𝑎) = ∩ ran 𝑈 → ∩ ran 𝑈 ≠ ∅))
36	6, 35	syl5bi 231	. . 3 ⊢ (𝑡:ω–1-1→𝑉 → (∩ ran 𝑈 ∈ ran 𝑈 → ∩ ran 𝑈 ≠ ∅))
37	36	adantl 481	. 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 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ∩ cint 4410   class class class wbr 4583  ran crn 5039  suc csuc 5642   Fn wfn 5799  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957  seq_𝜔cseqom 7429   ↑_𝑚 cmap 7744   ≈ cen 7838  Fincfn 7841

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-1o 7447  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845

This theorem is referenced by:  fin23lem31  9048