Theorem fin23lem16 9358

Description: Lemma for fin23 9412. 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 4603	. . 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 7702	. . . . 5 ⊢ 𝑈 Fn ω
4		fvelrnb 6385	. . . . 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 7231	. . . . . . . 8 ⊢ ∅ ∈ ω
7		0ss 4114	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑏
8	2	fin23lem15 9357	. . . . . . . . 9 ⊢ (((𝑏 ∈ ω ∧ ∅ ∈ ω) ∧ ∅ ⊆ 𝑏) → (𝑈‘𝑏) ⊆ (𝑈‘∅))
9	7, 8	mpan2 663	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ ∅ ∈ ω) → (𝑈‘𝑏) ⊆ (𝑈‘∅))
10	6, 9	mpan2 663	. . . . . . 7 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ (𝑈‘∅))
11		vex 3352	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
12	11	rnex 7246	. . . . . . . . 9 ⊢ ran 𝑡 ∈ V
13	12	uniex 7099	. . . . . . . 8 ⊢ ∪ ran 𝑡 ∈ V
14	2	seqom0g 7703	. . . . . . . 8 ⊢ (∪ ran 𝑡 ∈ V → (𝑈‘∅) = ∪ ran 𝑡)
15	13, 14	ax-mp 5	. . . . . . 7 ⊢ (𝑈‘∅) = ∪ ran 𝑡
16	10, 15	syl6sseq 3798	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ ∪ ran 𝑡)
17		sseq1 3773	. . . . . 6 ⊢ ((𝑈‘𝑏) = 𝑎 → ((𝑈‘𝑏) ⊆ ∪ ran 𝑡 ↔ 𝑎 ⊆ ∪ ran 𝑡))
18	16, 17	syl5ibcom 235	. . . . 5 ⊢ (𝑏 ∈ ω → ((𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡))
19	18	rexlimiv 3174	. . . 4 ⊢ (∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡)
20	5, 19	sylbi 207	. . 3 ⊢ (𝑎 ∈ ran 𝑈 → 𝑎 ⊆ ∪ ran 𝑡)
21	1, 20	mprgbir 3075	. 2 ⊢ ∪ ran 𝑈 ⊆ ∪ ran 𝑡
22		fnfvelrn 6499	. . . . 5 ⊢ ((𝑈 Fn ω ∧ ∅ ∈ ω) → (𝑈‘∅) ∈ ran 𝑈)
23	3, 6, 22	mp2an 664	. . . 4 ⊢ (𝑈‘∅) ∈ ran 𝑈
24	15, 23	eqeltrri 2846	. . 3 ⊢ ∪ ran 𝑡 ∈ ran 𝑈
25		elssuni 4601	. . 3 ⊢ (∪ ran 𝑡 ∈ ran 𝑈 → ∪ ran 𝑡 ⊆ ∪ ran 𝑈)
26	24, 25	ax-mp 5	. 2 ⊢ ∪ ran 𝑡 ⊆ ∪ ran 𝑈
27	21, 26	eqssi 3766	1 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡

Syntax hints:   ↔ wb 196   ∧ wa 382   = wceq 1630   ∈ wcel 2144  ∃wrex 3061  Vcvv 3349   ∩ cin 3720   ⊆ wss 3721  ∅c0 4061  ifcif 4223  ∪ cuni 4572  ran crn 5250   Fn wfn 6026  ‘cfv 6031   ↦ cmpt2 6794  ωcom 7211  seq_𝜔cseqom 7694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-seqom 7695

This theorem is referenced by:  fin23lem17  9361  fin23lem31  9366