Theorem fin23lem16 10091

Description: Lemma for fin23 10145. 𝑈 ranges over the original set; in particular ran 𝑈 is a set, although we do not assume here that 𝑈 is. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Hypothesis

Ref	Expression
fin23lem.a	⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)

Assertion

Ref	Expression
fin23lem16	⊢ ∪ ran 𝑈 = ∪ ran 𝑡

Distinct variable groups:   𝑡,𝑖,𝑢   𝑈,𝑖,𝑢

Allowed substitution hint:   𝑈(𝑡)

Proof of Theorem fin23lem16

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unissb 4873	. . 3 ⊢ (∪ ran 𝑈 ⊆ ∪ ran 𝑡 ↔ ∀𝑎 ∈ ran 𝑈 𝑎 ⊆ ∪ ran 𝑡)
2		fin23lem.a	. . . . . 6 ⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
3	2	fnseqom 8286	. . . . 5 ⊢ 𝑈 Fn ω
4		fvelrnb 6830	. . . . 5 ⊢ (𝑈 Fn ω → (𝑎 ∈ ran 𝑈 ↔ ∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎))
5	3, 4	ax-mp 5	. . . 4 ⊢ (𝑎 ∈ ran 𝑈 ↔ ∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎)
6		peano1 7735	. . . . . . . 8 ⊢ ∅ ∈ ω
7		0ss 4330	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑏
8	2	fin23lem15 10090	. . . . . . . . 9 ⊢ (((𝑏 ∈ ω ∧ ∅ ∈ ω) ∧ ∅ ⊆ 𝑏) → (𝑈‘𝑏) ⊆ (𝑈‘∅))
9	7, 8	mpan2 688	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ ∅ ∈ ω) → (𝑈‘𝑏) ⊆ (𝑈‘∅))
10	6, 9	mpan2 688	. . . . . . 7 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ (𝑈‘∅))
11		vex 3436	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
12	11	rnex 7759	. . . . . . . . 9 ⊢ ran 𝑡 ∈ V
13	12	uniex 7594	. . . . . . . 8 ⊢ ∪ ran 𝑡 ∈ V
14	2	seqom0g 8287	. . . . . . . 8 ⊢ (∪ ran 𝑡 ∈ V → (𝑈‘∅) = ∪ ran 𝑡)
15	13, 14	ax-mp 5	. . . . . . 7 ⊢ (𝑈‘∅) = ∪ ran 𝑡
16	10, 15	sseqtrdi 3971	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ ∪ ran 𝑡)
17		sseq1 3946	. . . . . 6 ⊢ ((𝑈‘𝑏) = 𝑎 → ((𝑈‘𝑏) ⊆ ∪ ran 𝑡 ↔ 𝑎 ⊆ ∪ ran 𝑡))
18	16, 17	syl5ibcom 244	. . . . 5 ⊢ (𝑏 ∈ ω → ((𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡))
19	18	rexlimiv 3209	. . . 4 ⊢ (∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡)
20	5, 19	sylbi 216	. . 3 ⊢ (𝑎 ∈ ran 𝑈 → 𝑎 ⊆ ∪ ran 𝑡)
21	1, 20	mprgbir 3079	. 2 ⊢ ∪ ran 𝑈 ⊆ ∪ ran 𝑡
22		fnfvelrn 6958	. . . . 5 ⊢ ((𝑈 Fn ω ∧ ∅ ∈ ω) → (𝑈‘∅) ∈ ran 𝑈)
23	3, 6, 22	mp2an 689	. . . 4 ⊢ (𝑈‘∅) ∈ ran 𝑈
24	15, 23	eqeltrri 2836	. . 3 ⊢ ∪ ran 𝑡 ∈ ran 𝑈
25		elssuni 4871	. . 3 ⊢ (∪ ran 𝑡 ∈ ran 𝑈 → ∪ ran 𝑡 ⊆ ∪ ran 𝑈)
26	24, 25	ax-mp 5	. 2 ⊢ ∪ ran 𝑡 ⊆ ∪ ran 𝑈
27	21, 26	eqssi 3937	1 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  ifcif 4459  ∪ cuni 4839  ran crn 5590   Fn wfn 6428  ‘cfv 6433   ∈ cmpo 7277  ωcom 7712  seq_ωcseqom 8278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-seqom 8279

This theorem is referenced by:  fin23lem17  10094  fin23lem31  10099