Theorem iunfi 9381

Description: The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 9382. Note that 𝐵 depends on 𝑥, i.e. can be thought of as 𝐵(𝑥). (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
iunfi	⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunfi

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 3321	. . . 4 ⊢ (𝑤 = ∅ → (∀𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ ∅ 𝐵 ∈ Fin))
2		iuneq1 5013	. . . . . 6 ⊢ (𝑤 = ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵)
3		0iun 5068	. . . . . 6 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅
4	2, 3	eqtrdi 2791	. . . . 5 ⊢ (𝑤 = ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 = ∅)
5	4	eleq1d 2824	. . . 4 ⊢ (𝑤 = ∅ → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∅ ∈ Fin))
6	1, 5	imbi12d 344	. . 3 ⊢ (𝑤 = ∅ → ((∀𝑥 ∈ 𝑤 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin) ↔ (∀𝑥 ∈ ∅ 𝐵 ∈ Fin → ∅ ∈ Fin)))
7		raleq 3321	. . . 4 ⊢ (𝑤 = 𝑦 → (∀𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ 𝑦 𝐵 ∈ Fin))
8		iuneq1 5013	. . . . 5 ⊢ (𝑤 = 𝑦 → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ 𝑦 𝐵)
9	8	eleq1d 2824	. . . 4 ⊢ (𝑤 = 𝑦 → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin))
10	7, 9	imbi12d 344	. . 3 ⊢ (𝑤 = 𝑦 → ((∀𝑥 ∈ 𝑤 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin) ↔ (∀𝑥 ∈ 𝑦 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin)))
11		raleq 3321	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
12		iuneq1 5013	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
13	12	eleq1d 2824	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
14	11, 13	imbi12d 344	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((∀𝑥 ∈ 𝑤 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin) ↔ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin)))
15		raleq 3321	. . . 4 ⊢ (𝑤 = 𝐴 → (∀𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin))
16		iuneq1 5013	. . . . 5 ⊢ (𝑤 = 𝐴 → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵)
17	16	eleq1d 2824	. . . 4 ⊢ (𝑤 = 𝐴 → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin))
18	15, 17	imbi12d 344	. . 3 ⊢ (𝑤 = 𝐴 → ((∀𝑥 ∈ 𝑤 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin) ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)))
19		0fi 9081	. . . 4 ⊢ ∅ ∈ Fin
20	19	a1i 11	. . 3 ⊢ (∀𝑥 ∈ ∅ 𝐵 ∈ Fin → ∅ ∈ Fin)
21		ssun1 4188	. . . . . . 7 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
22		ssralv 4064	. . . . . . 7 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ∀𝑥 ∈ 𝑦 𝐵 ∈ Fin))
23	21, 22	ax-mp 5	. . . . . 6 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ∀𝑥 ∈ 𝑦 𝐵 ∈ Fin)
24	23	imim1i 63	. . . . 5 ⊢ ((∀𝑥 ∈ 𝑦 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin))
25		iunxun 5099	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
26		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐵
27		nfcsb1v 3933	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
28		csbeq1a 3922	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
29	26, 27, 28	cbviun 5041	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ∪ 𝑦 ∈ {𝑧}⦋𝑦 / 𝑥⦌𝐵
30		vex 3482	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
31		csbeq1 3911	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
32	30, 31	iunxsn 5096	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ {𝑧}⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵
33	29, 32	eqtri 2763	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑥⦌𝐵
34		ssun2 4189	. . . . . . . . . . 11 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
35		vsnid 4668	. . . . . . . . . . 11 ⊢ 𝑧 ∈ {𝑧}
36	34, 35	sselii 3992	. . . . . . . . . 10 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
37		nfcsb1v 3933	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
38	37	nfel1 2920	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵 ∈ Fin
39		csbeq1a 3922	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
40	39	eleq1d 2824	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝐵 ∈ Fin ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ Fin))
41	38, 40	rspc 3610	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ⦋𝑧 / 𝑥⦌𝐵 ∈ Fin))
42	36, 41	ax-mp 5	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ⦋𝑧 / 𝑥⦌𝐵 ∈ Fin)
43	33, 42	eqeltrid 2843	. . . . . . . 8 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ∪ 𝑥 ∈ {𝑧}𝐵 ∈ Fin)
44		unfi 9210	. . . . . . . 8 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin ∧ ∪ 𝑥 ∈ {𝑧}𝐵 ∈ Fin) → (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵) ∈ Fin)
45	43, 44	sylan2 593	. . . . . . 7 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin) → (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵) ∈ Fin)
46	25, 45	eqeltrid 2843	. . . . . 6 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin) → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin)
47	46	expcom 413	. . . . 5 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → (∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
48	24, 47	sylcom 30	. . . 4 ⊢ ((∀𝑥 ∈ 𝑦 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
49	48	a1i 11	. . 3 ⊢ (𝑦 ∈ Fin → ((∀𝑥 ∈ 𝑦 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin)))
50	6, 10, 14, 18, 20, 49	findcard2 9203	. 2 ⊢ (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin))
51	50	imp 406	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ⦋csb 3908   ∪ cun 3961   ⊆ wss 3963  ∅c0 4339  {csn 4631  ∪ ciun 4996  Fincfn 8984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-en 8985  df-fin 8988

This theorem is referenced by:  unifi  9382  infssuni  9384  ixpfi  9387  ackbij1lem9  10265  ackbij1lem10  10266  fsuppmapnn0fiublem  14028  fsuppmapnn0fiub  14029  fsum2dlem  15803  fsumcom2  15807  fsumiun  15854  hashiun  15855  hash2iun  15856  ackbijnn  15861  fprod2dlem  16013  fprodcom2  16017  ablfaclem3  20122  pmatcoe1fsupp  22723  locfincmp  23550  txcmplem2  23666  alexsubALTlem3  24073  aannenlem1  26385  fsumvma  27272  numedglnl  29176  rabrexfi  32534  fsumiunle  32836  fedgmullem1  33657  poimirlem30  37637  fiphp3d  42807  hbt  43119  cnrefiisplem  45785