Theorem fin23lem31 10257

Description: Lemma for fin23 10303. 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 10244	. . 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 10255	. . . . 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 10253	. . . . . . 7 ⊢ ((∪ ran 𝑡 ∈ 𝐹 ∧ 𝑡:ω–1-1→𝑉) → ∩ ran 𝑈 ≠ ∅)
11	10	ancoms 459	. . . . . 6 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∩ ran 𝑈 ≠ ∅)
12		n0 4282	. . . . . 6 ⊢ (∩ ran 𝑈 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ ∩ ran 𝑈)
13	11, 12	sylib 219	. . . . 5 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∃𝑎 𝑎 ∈ ∩ ran 𝑈)
14	3	fnseqom 8385	. . . . . . . . . . . . 13 ⊢ 𝑈 Fn ω
15		fndm 6589	. . . . . . . . . . . . 13 ⊢ (𝑈 Fn ω → dom 𝑈 = ω)
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ dom 𝑈 = ω
17		peano1 7830	. . . . . . . . . . . . 13 ⊢ ∅ ∈ ω
18	17	ne0ii 4273	. . . . . . . . . . . 12 ⊢ ω ≠ ∅
19	16, 18	eqnetri 3004	. . . . . . . . . . 11 ⊢ dom 𝑈 ≠ ∅
20		dm0rn0 5867	. . . . . . . . . . . 12 ⊢ (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
21	20	necon3bii 2986	. . . . . . . . . . 11 ⊢ (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
22	19, 21	mpbi 231	. . . . . . . . . 10 ⊢ ran 𝑈 ≠ ∅
23		intssuni 4901	. . . . . . . . . 10 ⊢ (ran 𝑈 ≠ ∅ → ∩ ran 𝑈 ⊆ ∪ ran 𝑈)
24	22, 23	ax-mp 5	. . . . . . . . 9 ⊢ ∩ ran 𝑈 ⊆ ∪ ran 𝑈
25	3	fin23lem16 10249	. . . . . . . . 9 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡
26	24, 25	sseqtri 3963	. . . . . . . 8 ⊢ ∩ ran 𝑈 ⊆ ∪ ran 𝑡
27	26	sseli 3911	. . . . . . 7 ⊢ (𝑎 ∈ ∩ ran 𝑈 → 𝑎 ∈ ∪ ran 𝑡)
28		f1fun 6726	. . . . . . . . . . . . 13 ⊢ (𝑡:ω–1-1→𝑉 → Fun 𝑡)
29	28	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → Fun 𝑡)
30	3, 1, 4, 5, 6, 7	fin23lem30 10256	. . . . . . . . . . . 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 4379	. . . . . . . . . . 11 ⊢ ((∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ∈ ∪ ran 𝑍 ¬ 𝑎 ∈ ∩ ran 𝑈)
33	31, 32	sylib 219	. . . . . . . . . 10 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∀𝑎 ∈ ∪ ran 𝑍 ¬ 𝑎 ∈ ∩ ran 𝑈)
34		rsp 3227	. . . . . . . . . 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 407	. . . . . . 7 ⊢ (((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) ∧ 𝑎 ∈ ∩ ran 𝑈) → ¬ 𝑎 ∈ ∪ ran 𝑍)
38		nelne1 3031	. . . . . . 7 ⊢ ((𝑎 ∈ ∪ ran 𝑡 ∧ ¬ 𝑎 ∈ ∪ ran 𝑍) → ∪ ran 𝑡 ≠ ∪ ran 𝑍)
39	27, 37, 38	syl2an2 692	. . . . . 6 ⊢ (((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) ∧ 𝑎 ∈ ∩ ran 𝑈) → ∪ ran 𝑡 ≠ ∪ ran 𝑍)
40	39	necomd 2989	. . . . 5 ⊢ (((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) ∧ 𝑎 ∈ ∩ ran 𝑈) → ∪ ran 𝑍 ≠ ∪ ran 𝑡)
41	13, 40	exlimddv 1942	. . . 4 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∪ ran 𝑍 ≠ ∪ ran 𝑡)
42		df-pss 3903	. . . 4 ⊢ (∪ ran 𝑍 ⊊ ∪ ran 𝑡 ↔ (∪ ran 𝑍 ⊆ ∪ ran 𝑡 ∧ ∪ ran 𝑍 ≠ ∪ ran 𝑡))
43	9, 41, 42	sylanbrc 589	. . 3 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡)
44	2, 43	sylan2 599	. 2 ⊢ ((𝑡:ω–1-1→𝑉 ∧ (𝐺 ∈ 𝐹 ∧ ∪ ran 𝑡 ⊆ 𝐺)) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡)
45	44	3impb 1120	1 ⊢ ((𝑡:ω–1-1→𝑉 ∧ 𝐺 ∈ 𝐹 ∧ ∪ ran 𝑡 ⊆ 𝐺) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717   ≠ wne 2934  ∀wral 3053  {crab 3391  Vcvv 3431   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883   ⊊ wpss 3884  ∅c0 4262  ifcif 4455  𝒫 cpw 4530  ∪ cuni 4839  ∩ cint 4878   class class class wbr 5073   ↦ cmpt 5154  dom cdm 5619  ran crn 5620   ∘ ccom 5623  suc csuc 6313  Fun wfun 6480   Fn wfn 6481  –1-1→wf1 6483  ‘cfv 6486  ℩crio 7313  (class class class)co 7357   ∈ cmpo 7359  ωcom 7807  seq_ωcseqom 8377   ↑_m cmap 8764   ≈ cen 8881  Fincfn 8884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-int 4879  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-1st 7932  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-seqom 8378  df-1o 8396  df-er 8634  df-map 8766  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-card 9855

This theorem is referenced by:  fin23lem32  10258