Theorem fin23lem29 10366

Description: Lemma for fin23 10414. 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 4571	. . 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 215	. 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 5938	. . . . . 6 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ran 𝑍 = ran (𝑡 ∘ 𝑅))
5	4	unieqd 4922	. . . . 5 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ∪ ran 𝑍 = ∪ ran (𝑡 ∘ 𝑅))
6		rncoss 5975	. . . . . 6 ⊢ ran (𝑡 ∘ 𝑅) ⊆ ran 𝑡
7	6	unissi 4918	. . . . 5 ⊢ ∪ ran (𝑡 ∘ 𝑅) ⊆ ∪ ran 𝑡
8	5, 7	eqsstrdi 4031	. . . 4 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ∪ ran 𝑍 ⊆ ∪ ran 𝑡)
9	8	adantl 480	. . 3 ⊢ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) → ∪ ran 𝑍 ⊆ ∪ ran 𝑡)
10		rneq 5938	. . . . . 6 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
11	10	unieqd 4922	. . . . 5 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → ∪ ran 𝑍 = ∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
12		rncoss 5975	. . . . . . 7 ⊢ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
13	12	unissi 4918	. . . . . 6 ⊢ ∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
14		unissb 4943	. . . . . . 7 ⊢ (∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ⊆ ∪ ran 𝑡 ↔ ∀𝑎 ∈ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))𝑎 ⊆ ∪ ran 𝑡)
15		abid 2706	. . . . . . . . 9 ⊢ (𝑎 ∈ {𝑎 ∣ ∃𝑧 ∈ 𝑃 𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)} ↔ ∃𝑧 ∈ 𝑃 𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
16		fvssunirn 6929	. . . . . . . . . . . . 13 ⊢ (𝑡‘𝑧) ⊆ ∪ ran 𝑡
17	16	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑃 → (𝑡‘𝑧) ⊆ ∪ ran 𝑡)
18	17	ssdifssd 4139	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑃 → ((𝑡‘𝑧) ∖ ∩ ran 𝑈) ⊆ ∪ ran 𝑡)
19		sseq1 4002	. . . . . . . . . . 11 ⊢ (𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → (𝑎 ⊆ ∪ ran 𝑡 ↔ ((𝑡‘𝑧) ∖ ∩ ran 𝑈) ⊆ ∪ ran 𝑡))
20	18, 19	syl5ibrcom 246	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑃 → (𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → 𝑎 ⊆ ∪ ran 𝑡))
21	20	rexlimiv 3137	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝑃 𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → 𝑎 ⊆ ∪ ran 𝑡)
22	15, 21	sylbi 216	. . . . . . . 8 ⊢ (𝑎 ∈ {𝑎 ∣ ∃𝑧 ∈ 𝑃 𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)} → 𝑎 ⊆ ∪ ran 𝑡)
23		eqid 2725	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
24	23	rnmpt 5957	. . . . . . . 8 ⊢ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = {𝑎 ∣ ∃𝑧 ∈ 𝑃 𝑎 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)}
25	22, 24	eleq2s 2843	. . . . . . 7 ⊢ (𝑎 ∈ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) → 𝑎 ⊆ ∪ ran 𝑡)
26	14, 25	mprgbir 3057	. . . . . 6 ⊢ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ⊆ ∪ ran 𝑡
27	13, 26	sstri 3986	. . . . 5 ⊢ ∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ∪ ran 𝑡
28	11, 27	eqsstrdi 4031	. . . 4 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → ∪ ran 𝑍 ⊆ ∪ ran 𝑡)
29	28	adantl 480	. . 3 ⊢ ((¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) → ∪ ran 𝑍 ⊆ ∪ ran 𝑡)
30	9, 29	jaoi 855	. 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 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098  {cab 2702  ∀wral 3050  ∃wrex 3059  {crab 3418  Vcvv 3461   ∖ cdif 3941   ∩ cin 3943   ⊆ wss 3944  ∅c0 4322  ifcif 4530  𝒫 cpw 4604  ∪ cuni 4909  ∩ cint 4950   class class class wbr 5149   ↦ cmpt 5232  ran crn 5679   ∘ ccom 5682  suc csuc 6373  ‘cfv 6549  ℩crio 7374  (class class class)co 7419   ∈ cmpo 7421  ωcom 7871  seq_ωcseqom 8468   ↑_m cmap 8845   ≈ cen 8961  Fincfn 8964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fv 6557

This theorem is referenced by:  fin23lem31  10368