Theorem fin23lem31 10412

Description: Lemma for fin23 10458. The residual is has a strictly smaller range than the previous sequence. This will be iterated to build an unbounded chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)

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)}
fin23lem.b	⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)}
fin23lem.c	⊢ 𝑄 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ 𝑃 (𝑥 ∩ 𝑃) ≈ 𝑤))
fin23lem.d	⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
fin23lem.e	⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))

Assertion

Ref	Expression
fin23lem31	⊢ ((𝑡:ω–1-1→𝑉 ∧ 𝐺 ∈ 𝐹 ∧ ∪ ran 𝑡 ⊆ 𝐺) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡)

Distinct variable groups:   𝑔,𝑖,𝑡,𝑢,𝑣,𝑥,𝑧,𝑎   𝐹,𝑎,𝑡   𝑉,𝑎   𝑤,𝑎,𝑥,𝑧,𝑃   𝑣,𝑎,𝑅,𝑖,𝑢   𝑈,𝑎,𝑖,𝑢,𝑣,𝑧   𝑍,𝑎   𝑔,𝑎,𝐺,𝑡,𝑥

Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑔,𝑖)   𝑄(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖,𝑎)   𝑅(𝑥,𝑧,𝑤,𝑡,𝑔)   𝑈(𝑥,𝑤,𝑡,𝑔)   𝐹(𝑥,𝑧,𝑤,𝑣,𝑢,𝑔,𝑖)   𝐺(𝑧,𝑤,𝑣,𝑢,𝑖)   𝑉(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)   𝑍(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem31

Step	Hyp	Ref	Expression
1		fin23lem17.f	. . . 4 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
2	1	ssfin3ds 10399	. . 3 ⊢ ((𝐺 ∈ 𝐹 ∧ ∪ ran 𝑡 ⊆ 𝐺) → ∪ ran 𝑡 ∈ 𝐹)
3		fin23lem.a	. . . . . 6 ⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
4		fin23lem.b	. . . . . 6 ⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)}
5		fin23lem.c	. . . . . 6 ⊢ 𝑄 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ 𝑃 (𝑥 ∩ 𝑃) ≈ 𝑤))
6		fin23lem.d	. . . . . 6 ⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
7		fin23lem.e	. . . . . 6 ⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
8	3, 1, 4, 5, 6, 7	fin23lem29 10410	. . . . 5 ⊢ ∪ ran 𝑍 ⊆ ∪ ran 𝑡
9	8	a1i 11	. . . 4 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∪ ran 𝑍 ⊆ ∪ ran 𝑡)
10	3, 1	fin23lem21 10408	. . . . . . 7 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∩ ran 𝑈 ≠ ∅)
11	10	ancoms 458	. . . . . 6 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∩ ran 𝑈 ≠ ∅)
12		n0 4376	. . . . . 6 ⊢ (∩ ran 𝑈 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ ∩ ran 𝑈)
13	11, 12	sylib 218	. . . . 5 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∃𝑎 𝑎 ∈ ∩ ran 𝑈)
14	3	fnseqom 8511	. . . . . . . . . . . . 13 ⊢ 𝑈 Fn ω
15		fndm 6682	. . . . . . . . . . . . 13 ⊢ (𝑈 Fn ω → dom 𝑈 = ω)
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ dom 𝑈 = ω
17		peano1 7927	. . . . . . . . . . . . 13 ⊢ ∅ ∈ ω
18	17	ne0ii 4367	. . . . . . . . . . . 12 ⊢ ω ≠ ∅
19	16, 18	eqnetri 3017	. . . . . . . . . . 11 ⊢ dom 𝑈 ≠ ∅
20		dm0rn0 5949	. . . . . . . . . . . 12 ⊢ (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
21	20	necon3bii 2999	. . . . . . . . . . 11 ⊢ (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
22	19, 21	mpbi 230	. . . . . . . . . 10 ⊢ ran 𝑈 ≠ ∅
23		intssuni 4994	. . . . . . . . . 10 ⊢ (ran 𝑈 ≠ ∅ → ∩ ran 𝑈 ⊆ ∪ ran 𝑈)
24	22, 23	ax-mp 5	. . . . . . . . 9 ⊢ ∩ ran 𝑈 ⊆ ∪ ran 𝑈
25	3	fin23lem16 10404	. . . . . . . . 9 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡
26	24, 25	sseqtri 4045	. . . . . . . 8 ⊢ ∩ ran 𝑈 ⊆ ∪ ran 𝑡
27	26	sseli 4004	. . . . . . 7 ⊢ (𝑎 ∈ ∩ ran 𝑈 → 𝑎 ∈ ∪ ran 𝑡)
28		f1fun 6819	. . . . . . . . . . . . 13 ⊢ (𝑡:ω–1-1→𝑉 → Fun 𝑡)
29	28	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → Fun 𝑡)
30	3, 1, 4, 5, 6, 7	fin23lem30 10411	. . . . . . . . . . . 12 ⊢ (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅)
31	29, 30	syl 17	. . . . . . . . . . 11 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅)
32		disj 4473	. . . . . . . . . . 11 ⊢ ((∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ∈ ∪ ran 𝑍 ¬ 𝑎 ∈ ∩ ran 𝑈)
33	31, 32	sylib 218	. . . . . . . . . 10 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∀𝑎 ∈ ∪ ran 𝑍 ¬ 𝑎 ∈ ∩ ran 𝑈)
34		rsp 3253	. . . . . . . . . 10 ⊢ (∀𝑎 ∈ ∪ ran 𝑍 ¬ 𝑎 ∈ ∩ ran 𝑈 → (𝑎 ∈ ∪ ran 𝑍 → ¬ 𝑎 ∈ ∩ ran 𝑈))
35	33, 34	syl 17	. . . . . . . . 9 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → (𝑎 ∈ ∪ ran 𝑍 → ¬ 𝑎 ∈ ∩ ran 𝑈))
36	35	con2d 134	. . . . . . . 8 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → (𝑎 ∈ ∩ ran 𝑈 → ¬ 𝑎 ∈ ∪ ran 𝑍))
37	36	imp 406	. . . . . . 7 ⊢ (((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) ∧ 𝑎 ∈ ∩ ran 𝑈) → ¬ 𝑎 ∈ ∪ ran 𝑍)
38		nelne1 3045	. . . . . . 7 ⊢ ((𝑎 ∈ ∪ ran 𝑡 ∧ ¬ 𝑎 ∈ ∪ ran 𝑍) → ∪ ran 𝑡 ≠ ∪ ran 𝑍)
39	27, 37, 38	syl2an2 685	. . . . . 6 ⊢ (((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) ∧ 𝑎 ∈ ∩ ran 𝑈) → ∪ ran 𝑡 ≠ ∪ ran 𝑍)
40	39	necomd 3002	. . . . 5 ⊢ (((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) ∧ 𝑎 ∈ ∩ ran 𝑈) → ∪ ran 𝑍 ≠ ∪ ran 𝑡)
41	13, 40	exlimddv 1934	. . . 4 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∪ ran 𝑍 ≠ ∪ ran 𝑡)
42		df-pss 3996	. . . 4 ⊢ (∪ ran 𝑍 ⊊ ∪ ran 𝑡 ↔ (∪ ran 𝑍 ⊆ ∪ ran 𝑡 ∧ ∪ ran 𝑍 ≠ ∪ ran 𝑡))
43	9, 41, 42	sylanbrc 582	. . 3 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡)
44	2, 43	sylan2 592	. 2 ⊢ ((𝑡:ω–1-1→𝑉 ∧ (𝐺 ∈ 𝐹 ∧ ∪ ran 𝑡 ⊆ 𝐺)) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡)
45	44	3impb 1115	1 ⊢ ((𝑡:ω–1-1→𝑉 ∧ 𝐺 ∈ 𝐹 ∧ ∪ ran 𝑡 ⊆ 𝐺) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717   ≠ wne 2946  ∀wral 3067  {crab 3443  Vcvv 3488   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976   ⊊ wpss 3977  ∅c0 4352  ifcif 4548  𝒫 cpw 4622  ∪ cuni 4931  ∩ cint 4970   class class class wbr 5166   ↦ cmpt 5249  dom cdm 5700  ran crn 5701   ∘ ccom 5704  suc csuc 6397  Fun wfun 6567   Fn wfn 6568  –1-1→wf1 6570  ‘cfv 6573  ℩crio 7403  (class class class)co 7448   ∈ cmpo 7450  ωcom 7903  seq_ωcseqom 8503   ↑_m cmap 8884   ≈ cen 9000  Fincfn 9003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-seqom 8504  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008

This theorem is referenced by:  fin23lem32  10413