Theorem fin23lem16 10372

Description: Lemma for fin23 10426. 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 4943	. . 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 8493	. . . . 5 ⊢ 𝑈 Fn ω
4		fvelrnb 6968	. . . . 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 7910	. . . . . . . 8 ⊢ ∅ ∈ ω
7		0ss 4405	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑏
8	2	fin23lem15 10371	. . . . . . . . 9 ⊢ (((𝑏 ∈ ω ∧ ∅ ∈ ω) ∧ ∅ ⊆ 𝑏) → (𝑈‘𝑏) ⊆ (𝑈‘∅))
9	7, 8	mpan2 691	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ ∅ ∈ ω) → (𝑈‘𝑏) ⊆ (𝑈‘∅))
10	6, 9	mpan2 691	. . . . . . 7 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ (𝑈‘∅))
11		vex 3481	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
12	11	rnex 7932	. . . . . . . . 9 ⊢ ran 𝑡 ∈ V
13	12	uniex 7759	. . . . . . . 8 ⊢ ∪ ran 𝑡 ∈ V
14	2	seqom0g 8494	. . . . . . . 8 ⊢ (∪ ran 𝑡 ∈ V → (𝑈‘∅) = ∪ ran 𝑡)
15	13, 14	ax-mp 5	. . . . . . 7 ⊢ (𝑈‘∅) = ∪ ran 𝑡
16	10, 15	sseqtrdi 4045	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ ∪ ran 𝑡)
17		sseq1 4020	. . . . . 6 ⊢ ((𝑈‘𝑏) = 𝑎 → ((𝑈‘𝑏) ⊆ ∪ ran 𝑡 ↔ 𝑎 ⊆ ∪ ran 𝑡))
18	16, 17	syl5ibcom 245	. . . . 5 ⊢ (𝑏 ∈ ω → ((𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡))
19	18	rexlimiv 3145	. . . 4 ⊢ (∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡)
20	5, 19	sylbi 217	. . 3 ⊢ (𝑎 ∈ ran 𝑈 → 𝑎 ⊆ ∪ ran 𝑡)
21	1, 20	mprgbir 3065	. 2 ⊢ ∪ ran 𝑈 ⊆ ∪ ran 𝑡
22		fnfvelrn 7099	. . . . 5 ⊢ ((𝑈 Fn ω ∧ ∅ ∈ ω) → (𝑈‘∅) ∈ ran 𝑈)
23	3, 6, 22	mp2an 692	. . . 4 ⊢ (𝑈‘∅) ∈ ran 𝑈
24	15, 23	eqeltrri 2835	. . 3 ⊢ ∪ ran 𝑡 ∈ ran 𝑈
25		elssuni 4941	. . 3 ⊢ (∪ ran 𝑡 ∈ ran 𝑈 → ∪ ran 𝑡 ⊆ ∪ ran 𝑈)
26	24, 25	ax-mp 5	. 2 ⊢ ∪ ran 𝑡 ⊆ ∪ ran 𝑈
27	21, 26	eqssi 4011	1 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∃wrex 3067  Vcvv 3477   ∩ cin 3961   ⊆ wss 3962  ∅c0 4338  ifcif 4530  ∪ cuni 4911  ran crn 5689   Fn wfn 6557  ‘cfv 6562   ∈ cmpo 7432  ωcom 7886  seq_ωcseqom 8485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-seqom 8486

This theorem is referenced by:  fin23lem17  10375  fin23lem31  10380