Theorem fin23lem16 10452

Description: Lemma for fin23 10506. 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 4906	. . 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 8503	. . . . 5 ⊢ 𝑈 Fn ω
4		fvelrnb 6976	. . . . 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 7934	. . . . . . . 8 ⊢ ∅ ∈ ω
7		0ss 4353	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑏
8	2	fin23lem15 10451	. . . . . . . . 9 ⊢ (((𝑏 ∈ ω ∧ ∅ ∈ ω) ∧ ∅ ⊆ 𝑏) → (𝑈‘𝑏) ⊆ (𝑈‘∅))
9	7, 8	mpan2 704	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ ∅ ∈ ω) → (𝑈‘𝑏) ⊆ (𝑈‘∅))
10	6, 9	mpan2 704	. . . . . . 7 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ (𝑈‘∅))
11		vex 3458	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
12	11	rnex 7956	. . . . . . . . 9 ⊢ ran 𝑡 ∈ V
13	12	uniex 7790	. . . . . . . 8 ⊢ ∪ ran 𝑡 ∈ V
14	2	seqom0g 8504	. . . . . . . 8 ⊢ (∪ ran 𝑡 ∈ V → (𝑈‘∅) = ∪ ran 𝑡)
15	13, 14	ax-mp 5	. . . . . . 7 ⊢ (𝑈‘∅) = ∪ ran 𝑡
16	10, 15	sseqtrdi 3953	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ ∪ ran 𝑡)
17		sseq1 3938	. . . . . 6 ⊢ ((𝑈‘𝑏) = 𝑎 → ((𝑈‘𝑏) ⊆ ∪ ran 𝑡 ↔ 𝑎 ⊆ ∪ ran 𝑡))
18	16, 17	syl5ibcom 248	. . . . 5 ⊢ (𝑏 ∈ ω → ((𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡))
19	18	rexlimiv 3160	. . . 4 ⊢ (∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡)
20	5, 19	sylbi 220	. . 3 ⊢ (𝑎 ∈ ran 𝑈 → 𝑎 ⊆ ∪ ran 𝑡)
21	1, 20	mprgbir 3087	. 2 ⊢ ∪ ran 𝑈 ⊆ ∪ ran 𝑡
22		fnfvelrn 7112	. . . . 5 ⊢ ((𝑈 Fn ω ∧ ∅ ∈ ω) → (𝑈‘∅) ∈ ran 𝑈)
23	3, 6, 22	mp2an 705	. . . 4 ⊢ (𝑈‘∅) ∈ ran 𝑈
24	15, 23	eqeltrri 2861	. . 3 ⊢ ∪ ran 𝑡 ∈ ran 𝑈
25		elssuni 4904	. . 3 ⊢ (∪ ran 𝑡 ∈ ran 𝑈 → ∪ ran 𝑡 ⊆ ∪ ran 𝑈)
26	24, 25	ax-mp 5	. 2 ⊢ ∪ ran 𝑡 ⊆ ∪ ran 𝑈
27	21, 26	eqssi 3929	1 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  ifcif 4485  ∪ cuni 4872  ran crn 5659   Fn wfn 6561  ‘cfv 6566   ∈ cmpo 7453  ωcom 7911  seq_ωcseqom 8495

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-seqom 8496

This theorem is used by:  fin23lem17  10455  fin23lem31  10460