Theorem fin23lem29 10340

Description: Lemma for fin23 10388. The residual is built from the same elements as the previous sequence. (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
fin23lem29	⊢ ∪ ran 𝑍 ⊆ ∪ ran 𝑡

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

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)   𝑍(𝑥, 𝑧, 𝑤, 𝑣, 𝑢, 𝑡, 𝑔, 𝑖)

Proof of Theorem fin23lem29

Step	Hyp	Ref	Expression
1		fin23lem.e	. 2 ⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
2		eqif 4531	. . 3 ⊢ (𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) ↔ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))))
3	2	biimpi 219	. 2 ⊢ (𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) → ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))))
4		rneq 5928	. . . . . 6 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ran 𝑍 = ran (𝑡 ∘ 𝑅))
5	4	unieqd 4887	. . . . 5 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ∪ ran 𝑍 = ∪ ran (𝑡 ∘ 𝑅))
6		rncoss 5969	. . . . . 6 ⊢ ran (𝑡 ∘ 𝑅) ⊆ ran 𝑡
7	6	unissi 4883	. . . . 5 ⊢ ∪ ran (𝑡 ∘ 𝑅) ⊆ ∪ ran 𝑡
8	5, 7	eqsstrdi 3982	. . . 4 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ∪ ran 𝑍 ⊆ ∪ ran 𝑡)
9	8	adantl 487	. . 3 ⊢ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) → ∪ ran 𝑍 ⊆ ∪ ran 𝑡)
10		rneq 5928	. . . . . 6 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
11	10	unieqd 4887	. . . . 5 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → ∪ ran 𝑍 = ∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
12		rncoss 5969	. . . . . . 7 ⊢ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
13	12	unissi 4883	. . . . . 6 ⊢ ∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
14		unissb 4908	. . . . . . 7 ⊢ (∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ⊆ ∪ ran 𝑡 ↔ ∀𝑎 ∈ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))𝑎 ⊆ ∪ ran 𝑡)
15		abid 2747	. . . . . . . . 9 ⊢ (𝑎 ∈ {𝑎 ∣ ∃𝑧 ∈ 𝑃 𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)} ↔ ∃𝑧 ∈ 𝑃 𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
16		fvssunirn 6916	. . . . . . . . . . . . 13 ⊢ (𝑡‘𝑧) ⊆ ∪ ran 𝑡
17	16	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑃 → (𝑡‘𝑧) ⊆ ∪ ran 𝑡)
18	17	ssdifssd 4101	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑃 → ((𝑡‘𝑧) ∖ ∩ ran 𝑈) ⊆ ∪ ran 𝑡)
19		sseq1 3963	. . . . . . . . . . 11 ⊢ (𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → (𝑎 ⊆ ∪ ran 𝑡 ↔ ((𝑡‘𝑧) ∖ ∩ ran 𝑈) ⊆ ∪ ran 𝑡))
20	18, 19	syl5ibrcom 250	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑃 → (𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → 𝑎 ⊆ ∪ ran 𝑡))
21	20	rexlimiv 3161	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑃 𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → 𝑎 ⊆ ∪ ran 𝑡)
22	15, 21	sylbi 220	. . . . . . . 8 ⊢ (𝑎 ∈ {𝑎 ∣ ∃𝑧 ∈ 𝑃 𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)} → 𝑎 ⊆ ∪ ran 𝑡)
23		eqid 2765	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
24	23	rnmpt 5949	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = {𝑎 ∣ ∃𝑧 ∈ 𝑃 𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)}
25	22, 24	eleq2s 2883	. . . . . . 7 ⊢ (𝑎 ∈ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) → 𝑎 ⊆ ∪ ran 𝑡)
26	14, 25	mprgbir 3088	. . . . . 6 ⊢ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ⊆ ∪ ran 𝑡
27	13, 26	sstri 3947	. . . . 5 ⊢ ∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ∪ ran 𝑡
28	11, 27	eqsstrdi 3982	. . . 4 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → ∪ ran 𝑍 ⊆ ∪ ran 𝑡)
29	28	adantl 487	. . 3 ⊢ ((¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) → ∪ ran 𝑍 ⊆ ∪ ran 𝑡)
30	9, 29	jaoi 871	. 2 ⊢ (((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))) → ∪ ran 𝑍 ⊆ ∪ ran 𝑡)
31	1, 3, 30	mp2b 10	1 ⊢ ∪ ran 𝑍 ⊆ ∪ ran 𝑡

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146  {cab 2743  ∀wral 3081  ∃wrex 3091  {crab 3418  Vcvv 3457   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  ifcif 4489  𝒫 cpw 4564  ∪ cuni 4874  ∩ cint 4914   class class class wbr 5111   ↦ cmpt 5194  ran crn 5664   ∘ ccom 5667  suc csuc 6366  ‘cfv 6540  ℩crio 7375  (class class class)co 7419   ∈ cmpo 7421  ωcom 7868  seq_ωcseqom 8440   ↑_m cmap 8830   ≈ cen 8946  Fincfn 8949

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fv 6548

This theorem is used by:  fin23lem31  10342