Theorem iunfi 9342

Description: The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 9343. 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 3322	. . . 4 ⊢ (𝑤 = ∅ → (∀𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ ∅ 𝐵 ∈ Fin))
2		iuneq1 5013	. . . . . 6 ⊢ (𝑤 = ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵)
3		0iun 5066	. . . . . 6 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅
4	2, 3	eqtrdi 2788	. . . . 5 ⊢ (𝑤 = ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 = ∅)
5	4	eleq1d 2818	. . . 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 3322	. . . 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 2818	. . . 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 3322	. . . 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 2818	. . . 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 3322	. . . 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 2818	. . . 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		0fin 9173	. . . 4 ⊢ ∅ ∈ Fin
20	19	a1i 11	. . 3 ⊢ (∀𝑥 ∈ ∅ 𝐵 ∈ Fin → ∅ ∈ Fin)
21		ssun1 4172	. . . . . . 7 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
22		ssralv 4050	. . . . . . 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 5097	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
26		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐵
27		nfcsb1v 3918	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
28		csbeq1a 3907	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
29	26, 27, 28	cbviun 5039	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ∪ 𝑦 ∈ {𝑧}⦋𝑦 / 𝑥⦌𝐵
30		vex 3478	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
31		csbeq1 3896	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
32	30, 31	iunxsn 5094	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ {𝑧}⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵
33	29, 32	eqtri 2760	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑥⦌𝐵
34		ssun2 4173	. . . . . . . . . . 11 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
35		vsnid 4665	. . . . . . . . . . 11 ⊢ 𝑧 ∈ {𝑧}
36	34, 35	sselii 3979	. . . . . . . . . 10 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
37		nfcsb1v 3918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
38	37	nfel1 2919	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵 ∈ Fin
39		csbeq1a 3907	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
40	39	eleq1d 2818	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝐵 ∈ Fin ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ Fin))
41	38, 40	rspc 3600	. . . . . . . . . 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 2837	. . . . . . . 8 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ∪ 𝑥 ∈ {𝑧}𝐵 ∈ Fin)
44		unfi 9174	. . . . . . . 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 2837	. . . . . 6 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin) → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin)
47	46	expcom 414	. . . . 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 9166	. 2 ⊢ (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin))
51	50	imp 407	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ⦋csb 3893   ∪ cun 3946   ⊆ wss 3948  ∅c0 4322  {csn 4628  ∪ ciun 4997  Fincfn 8941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7858  df-en 8942  df-fin 8945

This theorem is referenced by:  unifi  9343  infssuni  9345  ixpfi  9351  ackbij1lem9  10225  ackbij1lem10  10226  fsuppmapnn0fiublem  13959  fsuppmapnn0fiub  13960  fsum2dlem  15720  fsumcom2  15724  fsumiun  15771  hashiun  15772  hash2iun  15773  ackbijnn  15778  fprod2dlem  15928  fprodcom2  15932  ablfaclem3  19998  pmatcoe1fsupp  22423  locfincmp  23250  txcmplem2  23366  alexsubALTlem3  23773  aannenlem1  26065  fsumvma  26940  numedglnl  28659  fsumiunle  32290  fedgmullem1  32990  poimirlem30  36821  fiphp3d  41859  hbt  42174  cnrefiisplem  44844