Theorem iunfi 9332

Description: The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 9333. 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 4967	. . . . . 6 ⊢ (𝑤 = ∅ → ∪ 𝑥 ∈ 𝑤 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵)
3		0iun 5020	. . . . . 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 4967	. . . . 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 4967	. . . . 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 4967	. . . . 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 9070	. . . 4 ⊢ ∅ ∈ Fin
20	19	a1i 11	. . 3 ⊢ (∀𝑥 ∈ ∅ 𝐵 ∈ Fin → ∅ ∈ Fin)
21		ssun1 4176	. . . . . . 7 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
22		ssralv 3977	. . . . . . 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 5053	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = (∪ 𝑥 ∈ 𝑦 𝐵 ∪ ∪ 𝑥 ∈ {𝑧}𝐵)
26		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐵
27		nfcsb1v 3870	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
28		csbeq1a 3860	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
29	26, 27, 28	cbviun 4992	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ∪ 𝑦 ∈ {𝑧}⦋𝑦 / 𝑥⦌𝐵
30		vex 3454	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
31		csbeq1 3849	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
32	30, 31	iunxsn 5050	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ {𝑧}⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵
33	29, 32	eqtri 2783	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ {𝑧}𝐵 = ⦋𝑧 / 𝑥⦌𝐵
34		ssun2 4177	. . . . . . . . . . 11 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
35		vsnid 4623	. . . . . . . . . . 11 ⊢ 𝑧 ∈ {𝑧}
36	34, 35	sselii 3905	. . . . . . . . . 10 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
37		nfcsb1v 3870	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
38	37	nfel1 2938	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵 ∈ Fin
39		csbeq1a 3860	. . . . . . . . . . . 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 9186	. . . . . . . 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 9180	. 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 1569   ∈ wcel 2144  ∀wral 3076  ⦋csb 3846   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  {csn 4583  ∪ ciun 4950  Fincfn 8973

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-en 8974  df-fin 8977

This theorem is used by:  unifi  9333  infssuni  9335  ixpfi  9338  ackbij1lem9  10305  ackbij1lem10  10306  fsuppmapnn0fiublem  14133  fsuppmapnn0fiub  14134  fsum2dlem  15936  fsumcom2  15940  fsumiun  15988  hashiun  15989  hash2iun  15990  ackbijnn  15997  fprod2dlem  16147  fprodcom2  16151  chnfi  18808  ablfaclem3  20303  pmatcoe1fsupp  23019  locfincmp  23845  txcmplem2  23961  alexsubALTlem3  24368  rnplynfin  26630  aannenlem1  26655  fsumvma  27540  numedglnl  29722  rabrexfi  33102  fsumiunle  33420  psrmonprod  34184  fedgmullem1  34261  fldextrspunlsplem  34305  poimirlem30  38569  fiphp3d  43826  hbt  44132  cnrefiisplem  46839  tmachlem-franscan  47959