Theorem fin23lem31 10381

Description: Lemma for fin23 10427. 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 10368	. . 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 10379	. . . . 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 10377	. . . . . . 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 4359	. . . . . 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 8494	. . . . . . . . . . . . 13 ⊢ 𝑈 Fn ω
15		fndm 6672	. . . . . . . . . . . . 13 ⊢ (𝑈 Fn ω → dom 𝑈 = ω)
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ dom 𝑈 = ω
17		peano1 7911	. . . . . . . . . . . . 13 ⊢ ∅ ∈ ω
18	17	ne0ii 4350	. . . . . . . . . . . 12 ⊢ ω ≠ ∅
19	16, 18	eqnetri 3009	. . . . . . . . . . 11 ⊢ dom 𝑈 ≠ ∅
20		dm0rn0 5938	. . . . . . . . . . . 12 ⊢ (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
21	20	necon3bii 2991	. . . . . . . . . . 11 ⊢ (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
22	19, 21	mpbi 230	. . . . . . . . . 10 ⊢ ran 𝑈 ≠ ∅
23		intssuni 4975	. . . . . . . . . 10 ⊢ (ran 𝑈 ≠ ∅ → ∩ ran 𝑈 ⊆ ∪ ran 𝑈)
24	22, 23	ax-mp 5	. . . . . . . . 9 ⊢ ∩ ran 𝑈 ⊆ ∪ ran 𝑈
25	3	fin23lem16 10373	. . . . . . . . 9 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡
26	24, 25	sseqtri 4032	. . . . . . . 8 ⊢ ∩ ran 𝑈 ⊆ ∪ ran 𝑡
27	26	sseli 3991	. . . . . . 7 ⊢ (𝑎 ∈ ∩ ran 𝑈 → 𝑎 ∈ ∪ ran 𝑡)
28		f1fun 6807	. . . . . . . . . . . . 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 10380	. . . . . . . . . . . 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 4456	. . . . . . . . . . 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 3245	. . . . . . . . . 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 3037	. . . . . . 7 ⊢ ((𝑎 ∈ ∪ ran 𝑡 ∧ ¬ 𝑎 ∈ ∪ ran 𝑍) → ∪ ran 𝑡 ≠ ∪ ran 𝑍)
39	27, 37, 38	syl2an2 686	. . . . . 6 ⊢ (((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) ∧ 𝑎 ∈ ∩ ran 𝑈) → ∪ ran 𝑡 ≠ ∪ ran 𝑍)
40	39	necomd 2994	. . . . 5 ⊢ (((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) ∧ 𝑎 ∈ ∩ ran 𝑈) → ∪ ran 𝑍 ≠ ∪ ran 𝑡)
41	13, 40	exlimddv 1933	. . . 4 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∪ ran 𝑍 ≠ ∪ ran 𝑡)
42		df-pss 3983	. . . 4 ⊢ (∪ ran 𝑍 ⊊ ∪ ran 𝑡 ↔ (∪ ran 𝑍 ⊆ ∪ ran 𝑡 ∧ ∪ ran 𝑍 ≠ ∪ ran 𝑡))
43	9, 41, 42	sylanbrc 583	. . 3 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡)
44	2, 43	sylan2 593	. 2 ⊢ ((𝑡:ω–1-1→𝑉 ∧ (𝐺 ∈ 𝐹 ∧ ∪ ran 𝑡 ⊆ 𝐺)) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡)
45	44	3impb 1114	1 ⊢ ((𝑡:ω–1-1→𝑉 ∧ 𝐺 ∈ 𝐹 ∧ ∪ ran 𝑡 ⊆ 𝐺) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106  {cab 2712   ≠ wne 2938  ∀wral 3059  {crab 3433  Vcvv 3478   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963   ⊊ wpss 3964  ∅c0 4339  ifcif 4531  𝒫 cpw 4605  ∪ cuni 4912  ∩ cint 4951   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5689  ran crn 5690   ∘ ccom 5693  suc csuc 6388  Fun wfun 6557   Fn wfn 6558  –1-1→wf1 6560  ‘cfv 6563  ℩crio 7387  (class class class)co 7431   ∈ cmpo 7433  ωcom 7887  seq_ωcseqom 8486   ↑_m cmap 8865   ≈ cen 8981  Fincfn 8984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977

This theorem is referenced by:  fin23lem32  10382