Theorem fin23lem29 10254

Description: Lemma for fin23 10302. 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 4496	. . 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 217	. 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 5878	. . . . . 6 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ran 𝑍 = ran (𝑡 ∘ 𝑅))
5	4	unieqd 4851	. . . . 5 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ∪ ran 𝑍 = ∪ ran (𝑡 ∘ 𝑅))
6		rncoss 5919	. . . . . 6 ⊢ ran (𝑡 ∘ 𝑅) ⊆ ran 𝑡
7	6	unissi 4847	. . . . 5 ⊢ ∪ ran (𝑡 ∘ 𝑅) ⊆ ∪ ran 𝑡
8	5, 7	eqsstrdi 3959	. . . 4 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ∪ ran 𝑍 ⊆ ∪ ran 𝑡)
9	8	adantl 482	. . 3 ⊢ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) → ∪ ran 𝑍 ⊆ ∪ ran 𝑡)
10		rneq 5878	. . . . . 6 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
11	10	unieqd 4851	. . . . 5 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → ∪ ran 𝑍 = ∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
12		rncoss 5919	. . . . . . 7 ⊢ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
13	12	unissi 4847	. . . . . 6 ⊢ ∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
14		unissb 4871	. . . . . . 7 ⊢ (∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ⊆ ∪ ran 𝑡 ↔ ∀𝑎 ∈ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))𝑎 ⊆ ∪ ran 𝑡)
15		abid 2721	. . . . . . . . 9 ⊢ (𝑎 ∈ {𝑎 ∣ ∃𝑧 ∈ 𝑃 𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)} ↔ ∃𝑧 ∈ 𝑃 𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
16		fvssunirn 6858	. . . . . . . . . . . . 13 ⊢ (𝑡‘𝑧) ⊆ ∪ ran 𝑡
17	16	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑃 → (𝑡‘𝑧) ⊆ ∪ ran 𝑡)
18	17	ssdifssd 4077	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑃 → ((𝑡‘𝑧) ∖ ∩ ran 𝑈) ⊆ ∪ ran 𝑡)
19		sseq1 3940	. . . . . . . . . . 11 ⊢ (𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → (𝑎 ⊆ ∪ ran 𝑡 ↔ ((𝑡‘𝑧) ∖ ∩ ran 𝑈) ⊆ ∪ ran 𝑡))
20	18, 19	syl5ibrcom 248	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑃 → (𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → 𝑎 ⊆ ∪ ran 𝑡))
21	20	rexlimiv 3133	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑃 𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → 𝑎 ⊆ ∪ ran 𝑡)
22	15, 21	sylbi 218	. . . . . . . 8 ⊢ (𝑎 ∈ {𝑎 ∣ ∃𝑧 ∈ 𝑃 𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)} → 𝑎 ⊆ ∪ ran 𝑡)
23		eqid 2739	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
24	23	rnmpt 5899	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = {𝑎 ∣ ∃𝑧 ∈ 𝑃 𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)}
25	22, 24	eleq2s 2857	. . . . . . 7 ⊢ (𝑎 ∈ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) → 𝑎 ⊆ ∪ ran 𝑡)
26	14, 25	mprgbir 3060	. . . . . 6 ⊢ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ⊆ ∪ ran 𝑡
27	13, 26	sstri 3924	. . . . 5 ⊢ ∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ∪ ran 𝑡
28	11, 27	eqsstrdi 3959	. . . 4 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → ∪ ran 𝑍 ⊆ ∪ ran 𝑡)
29	28	adantl 482	. . 3 ⊢ ((¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) → ∪ ran 𝑍 ⊆ ∪ ran 𝑡)
30	9, 29	jaoi 863	. 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

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  {cab 2717  ∀wral 3053  ∃wrex 3063  {crab 3391  Vcvv 3431   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  ifcif 4454  𝒫 cpw 4529  ∪ cuni 4838  ∩ cint 4877   class class class wbr 5072   ↦ cmpt 5153  ran crn 5619   ∘ ccom 5622  suc csuc 6312  ‘cfv 6485  ℩crio 7312  (class class class)co 7356   ∈ cmpo 7358  ωcom 7806  seq_ωcseqom 8376   ↑_m cmap 8763   ≈ cen 8880  Fincfn 8883

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-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  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-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-iota 6441  df-fv 6493

This theorem is referenced by:  fin23lem31  10256