Theorem fin23lem21 10251

Description: Lemma for fin23 10301. 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 10250	. 2 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∩ ran 𝑈 ∈ ran 𝑈)
4	1	fnseqom 8386	. . . . 5 ⊢ 𝑈 Fn ω
5		fvelrnb 6894	. . . . 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 3444	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
9		f1f1orn 6785	. . . . . . . . . 10 ⊢ (𝑡:ω–1-1→𝑉 → 𝑡:ω–1-1-onto→ran 𝑡)
10		f1oen3g 8905	. . . . . . . . . 10 ⊢ ((𝑡 ∈ V ∧ 𝑡:ω–1-1-onto→ran 𝑡) → ω ≈ ran 𝑡)
11	8, 9, 10	sylancr 587	. . . . . . . . 9 ⊢ (𝑡:ω–1-1→𝑉 → ω ≈ ran 𝑡)
12		ominf 9166	. . . . . . . . 9 ⊢ ¬ ω ∈ Fin
13		ssdif0 4318	. . . . . . . . . . 11 ⊢ (ran 𝑡 ⊆ {∅} ↔ (ran 𝑡 ∖ {∅}) = ∅)
14		snfi 8982	. . . . . . . . . . . . 13 ⊢ {∅} ∈ Fin
15		ssfi 9099	. . . . . . . . . . . . 13 ⊢ (({∅} ∈ Fin ∧ ran 𝑡 ⊆ {∅}) → ran 𝑡 ∈ Fin)
16	14, 15	mpan 690	. . . . . . . . . . . 12 ⊢ (ran 𝑡 ⊆ {∅} → ran 𝑡 ∈ Fin)
17		enfi 9113	. . . . . . . . . . . 12 ⊢ (ω ≈ ran 𝑡 → (ω ∈ Fin ↔ ran 𝑡 ∈ Fin))
18	16, 17	imbitrrid 246	. . . . . . . . . . 11 ⊢ (ω ≈ ran 𝑡 → (ran 𝑡 ⊆ {∅} → ω ∈ Fin))
19	13, 18	biimtrrid 243	. . . . . . . . . 10 ⊢ (ω ≈ ran 𝑡 → ((ran 𝑡 ∖ {∅}) = ∅ → ω ∈ Fin))
20	19	necon3bd 2946	. . . . . . . . 9 ⊢ (ω ≈ ran 𝑡 → (¬ ω ∈ Fin → (ran 𝑡 ∖ {∅}) ≠ ∅))
21	11, 12, 20	mpisyl 21	. . . . . . . 8 ⊢ (𝑡:ω–1-1→𝑉 → (ran 𝑡 ∖ {∅}) ≠ ∅)
22		n0 4305	. . . . . . . . 9 ⊢ ((ran 𝑡 ∖ {∅}) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}))
23		eldifsn 4742	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (ran 𝑡 ∖ {∅}) ↔ (𝑎 ∈ ran 𝑡 ∧ 𝑎 ≠ ∅))
24		elssuni 4894	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ran 𝑡 → 𝑎 ⊆ ∪ ran 𝑡)
25		ssn0 4356	. . . . . . . . . . . 12 ⊢ ((𝑎 ⊆ ∪ ran 𝑡 ∧ 𝑎 ≠ ∅) → ∪ ran 𝑡 ≠ ∅)
26	24, 25	sylan 580	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ ran 𝑡 ∧ 𝑎 ≠ ∅) → ∪ ran 𝑡 ≠ ∅)
27	23, 26	sylbi 217	. . . . . . . . . 10 ⊢ (𝑎 ∈ (ran 𝑡 ∖ {∅}) → ∪ ran 𝑡 ≠ ∅)
28	27	exlimiv 1931	. . . . . . . . 9 ⊢ (∃𝑎 𝑎 ∈ (ran 𝑡 ∖ {∅}) → ∪ ran 𝑡 ≠ ∅)
29	22, 28	sylbi 217	. . . . . . . 8 ⊢ ((ran 𝑡 ∖ {∅}) ≠ ∅ → ∪ ran 𝑡 ≠ ∅)
30	21, 29	syl 17	. . . . . . 7 ⊢ (𝑡:ω–1-1→𝑉 → ∪ ran 𝑡 ≠ ∅)
31	1	fin23lem14 10245	. . . . . . 7 ⊢ ((𝑎 ∈ ω ∧ ∪ ran 𝑡 ≠ ∅) → (𝑈‘𝑎) ≠ ∅)
32	7, 30, 31	syl2anr 597	. . . . . 6 ⊢ ((𝑡:ω–1-1→𝑉 ∧ 𝑎 ∈ ω) → (𝑈‘𝑎) ≠ ∅)
33		neeq1 2994	. . . . . 6 ⊢ ((𝑈‘𝑎) = ∩ ran 𝑈 → ((𝑈‘𝑎) ≠ ∅ ↔ ∩ ran 𝑈 ≠ ∅))
34	32, 33	syl5ibcom 245	. . . . 5 ⊢ ((𝑡:ω–1-1→𝑉 ∧ 𝑎 ∈ ω) → ((𝑈‘𝑎) = ∩ ran 𝑈 → ∩ ran 𝑈 ≠ ∅))
35	34	rexlimdva 3137	. . . 4 ⊢ (𝑡:ω–1-1→𝑉 → (∃𝑎 ∈ ω (𝑈‘𝑎) = ∩ ran 𝑈 → ∩ ran 𝑈 ≠ ∅))
36	6, 35	biimtrid 242	. . 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 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  ifcif 4479  𝒫 cpw 4554  {csn 4580  ∪ cuni 4863  ∩ cint 4902   class class class wbr 5098  ran crn 5625  suc csuc 6319   Fn wfn 6487  –1-1→wf1 6489  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  ωcom 7808  seq_ωcseqom 8378   ↑_m cmap 8765   ≈ cen 8882  Fincfn 8885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-seqom 8379  df-1o 8397  df-map 8767  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889

This theorem is referenced by:  fin23lem31  10255