Theorem iunfi 9310

Description: The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 9311. 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 3316	. . . 4 ⊢ (𝑤 = ∅ → (∀𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ ∅ 𝐵 ∈ Fin))
2		iuneq1 4968	. . . . . 6 ⊢ (𝑤 = ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵)
3		0iun 5021	. . . . . 6 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅
4	2, 3	eqtrdi 2811	. . . . 5 ⊢ (𝑤 = ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 = ∅)
5	4	eleq1d 2845	. . . 4 ⊢ (𝑤 = ∅ → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∅ ∈ Fin))
6	1, 5	imbi12d 347	. . 3 ⊢ (𝑤 = ∅ → ((∀𝑥 ∈ 𝑤 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin) ↔ (∀𝑥 ∈ ∅ 𝐵 ∈ Fin → ∅ ∈ Fin)))
7		raleq 3316	. . . 4 ⊢ (𝑤 = 𝑦 → (∀𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ 𝑦 𝐵 ∈ Fin))
8		iuneq1 4968	. . . . 5 ⊢ (𝑤 = 𝑦 → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ 𝑦 𝐵)
9	8	eleq1d 2845	. . . 4 ⊢ (𝑤 = 𝑦 → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin))
10	7, 9	imbi12d 347	. . 3 ⊢ (𝑤 = 𝑦 → ((∀𝑥 ∈ 𝑤 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin) ↔ (∀𝑥 ∈ 𝑦 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin)))
11		raleq 3316	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
12		iuneq1 4968	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
13	12	eleq1d 2845	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
14	11, 13	imbi12d 347	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((∀𝑥 ∈ 𝑤 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin) ↔ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin)))
15		raleq 3316	. . . 4 ⊢ (𝑤 = 𝐴 → (∀𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin))
16		iuneq1 4968	. . . . 5 ⊢ (𝑤 = 𝐴 → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ 𝐴 𝐵)
17	16	eleq1d 2845	. . . 4 ⊢ (𝑤 = 𝐴 → (∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin))
18	15, 17	imbi12d 347	. . 3 ⊢ (𝑤 = 𝐴 → ((∀𝑥 ∈ 𝑤 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑤 𝐵 ∈ Fin) ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)))
19		0fi 9049	. . . 4 ⊢ ∅ ∈ Fin
20	19	a1i 11	. . 3 ⊢ (∀𝑥 ∈ ∅ 𝐵 ∈ Fin → ∅ ∈ Fin)
21		ssun1 4124	. . . . . . 7 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
22		ssralv 4000	. . . . . . 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 64	. . . . 5 ⊢ ((∀𝑥 ∈ 𝑦 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin) → (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin))
25		iunxun 5054	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
26		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐵
27		nfcsb1v 3871	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
28		csbeq1a 3861	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
29	26, 27, 28	cbviun 4993	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ∪ 𝑦 ∈ {𝑧}⦋𝑦 / 𝑥⦌𝐵
30		vex 3454	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
31		csbeq1 3850	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
32	30, 31	iunxsn 5051	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ {𝑧}⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵
33	29, 32	eqtri 2783	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑥⦌𝐵
34		ssun2 4125	. . . . . . . . . . 11 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
35		vsnid 4624	. . . . . . . . . . 11 ⊢ 𝑧 ∈ {𝑧}
36	34, 35	sselii 3928	. . . . . . . . . 10 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
37		nfcsb1v 3871	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
38	37	nfel1 2938	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵 ∈ Fin
39		csbeq1a 3861	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
40	39	eleq1d 2845	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝐵 ∈ Fin ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ Fin))
41	38, 40	rspc 3564	. . . . . . . . . 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 2864	. . . . . . . 8 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → ∪ 𝑥 ∈ {𝑧}𝐵 ∈ Fin)
44		unfi 9165	. . . . . . . 8 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin ∧ ∪ 𝑥 ∈ {𝑧}𝐵 ∈ Fin) → (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵) ∈ Fin)
45	43, 44	sylan2 605	. . . . . . 7 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin) → (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵) ∈ Fin)
46	25, 45	eqeltrid 2864	. . . . . 6 ⊢ ((∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin) → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin)
47	46	expcom 419	. . . . 5 ⊢ (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin → (∪ 𝑥 ∈ 𝑦 𝐵 ∈ Fin → ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ Fin))
48	24, 47	sylcom 31	. . . 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 9159	. 2 ⊢ (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 𝐵 ∈ Fin → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin))
51	50	imp 412	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ Fin) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ Fin)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ⦋csb 3847   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279  {csn 4584  ∪ ciun 4951  Fincfn 8952

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7735

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-om 7862  df-en 8953  df-fin 8956

This theorem is used by:  unifi  9311  infssuni  9313  ixpfi  9316  ackbij1lem9  10262  ackbij1lem10  10263  fsuppmapnn0fiublem  14087  fsuppmapnn0fiub  14088  fsum2dlem  15889  fsumcom2  15893  fsumiun  15941  hashiun  15942  hash2iun  15943  ackbijnn  15950  fprod2dlem  16100  fprodcom2  16104  chnfi  18755  ablfaclem3  20250  pmatcoe1fsupp  22966  locfincmp  23792  txcmplem2  23908  alexsubALTlem3  24315  rnplynfin  26579  aannenlem1  26604  fsumvma  27489  numedglnl  29641  rabrexfi  33021  fsumiunle  33339  psrmonprod  34103  fedgmullem1  34180  fldextrspunlsplem  34224  poimirlem30  38482  fiphp3d  43758  hbt  44069  cnrefiisplem  46755  tmachlem-franscan  47875