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 458	. . . . . 6 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∩ ran 𝑈 ≠ ∅)
12		n0 4306	. . . . . 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 8388	. . . . . . . . . . . . 13 ⊢ 𝑈 Fn ω
15		fndm 6596	. . . . . . . . . . . . 13 ⊢ (𝑈 Fn ω → dom 𝑈 = ω)
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ dom 𝑈 = ω
17		peano1 7833	. . . . . . . . . . . . 13 ⊢ ∅ ∈ ω
18	17	ne0ii 4297	. . . . . . . . . . . 12 ⊢ ω ≠ ∅
19	16, 18	eqnetri 3003	. . . . . . . . . . 11 ⊢ dom 𝑈 ≠ ∅
20		dm0rn0 5874	. . . . . . . . . . . 12 ⊢ (dom 𝑈 = ∅ ↔ ran 𝑈 = ∅)
21	20	necon3bii 2985	. . . . . . . . . . 11 ⊢ (dom 𝑈 ≠ ∅ ↔ ran 𝑈 ≠ ∅)
22	19, 21	mpbi 230	. . . . . . . . . 10 ⊢ ran 𝑈 ≠ ∅
23		intssuni 4926	. . . . . . . . . 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 3983	. . . . . . . 8 ⊢ ∩ ran 𝑈 ⊆ ∪ ran 𝑡
27	26	sseli 3930	. . . . . . 7 ⊢ (𝑎 ∈ ∩ ran 𝑈 → 𝑎 ∈ ∪ ran 𝑡)
28		f1fun 6733	. . . . . . . . . . . . 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 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 4403	. . . . . . . . . . 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 3225	. . . . . . . . . 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 3030	. . . . . . 7 ⊢ ((𝑎 ∈ ∪ ran 𝑡 ∧ ¬ 𝑎 ∈ ∪ ran 𝑍) → ∪ ran 𝑡 ≠ ∪ ran 𝑍)
39	27, 37, 38	syl2an2 687	. . . . . 6 ⊢ (((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) ∧ 𝑎 ∈ ∩ ran 𝑈) → ∪ ran 𝑡 ≠ ∪ ran 𝑍)
40	39	necomd 2988	. . . . 5 ⊢ (((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) ∧ 𝑎 ∈ ∩ ran 𝑈) → ∪ ran 𝑍 ≠ ∪ ran 𝑡)
41	13, 40	exlimddv 1937	. . . 4 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∪ ran 𝑍 ≠ ∪ ran 𝑡)
42		df-pss 3922	. . . 4 ⊢ (∪ ran 𝑍 ⊊ ∪ ran 𝑡 ↔ (∪ ran 𝑍 ⊆ ∪ ran 𝑡 ∧ ∪ ran 𝑍 ≠ ∪ ran 𝑡))
43	9, 41, 42	sylanbrc 584	. . 3 ⊢ ((𝑡:ω–1-1→𝑉 ∧ ∪ ran 𝑡 ∈ 𝐹) → ∪ ran 𝑍 ⊊ ∪ ran 𝑡)
44	2, 43	sylan2 594	. 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 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  {crab 3400  Vcvv 3441   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902   ⊊ wpss 3903  ∅c0 4286  ifcif 4480  𝒫 cpw 4555  ∪ cuni 4864  ∩ cint 4903   class class class wbr 5099   ↦ cmpt 5180  dom cdm 5625  ran crn 5626   ∘ ccom 5629  suc csuc 6320  Fun wfun 6487   Fn wfn 6488  –1-1→wf1 6490  ‘cfv 6493  ℩crio 7316  (class class class)co 7360   ∈ cmpo 7362  ωcom 7810  seq_ωcseqom 8380   ↑_m cmap 8767   ≈ cen 8884  Fincfn 8887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-seqom 8381  df-1o 8399  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-card 9855

This theorem is referenced by:  fin23lem32  10258