Theorem xpfi 7002

Description: The Cartesian product of two finite sets is finite. Lemma 8.1.16 of [AczelRathjen], p. 74. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Mar-2015.)

Assertion

Ref	Expression
xpfi	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)

Proof of Theorem xpfi

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

Step	Hyp	Ref	Expression
1		xpeq1 4678	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 × 𝐵) = (∅ × 𝐵))
2	1	eleq1d 2265	. . . 4 ⊢ (𝑥 = ∅ → ((𝑥 × 𝐵) ∈ Fin ↔ (∅ × 𝐵) ∈ Fin))
3	2	imbi2d 230	. . 3 ⊢ (𝑥 = ∅ → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → (∅ × 𝐵) ∈ Fin)))
4		xpeq1 4678	. . . . 5 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝑥 × 𝐵) = ((𝑦 ∖ {𝑧}) × 𝐵))
5	4	eleq1d 2265	. . . 4 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → ((𝑥 × 𝐵) ∈ Fin ↔ ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin))
6	5	imbi2d 230	. . 3 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin)))
7		xpeq1 4678	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 × 𝐵) = (𝑦 × 𝐵))
8	7	eleq1d 2265	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 × 𝐵) ∈ Fin ↔ (𝑦 × 𝐵) ∈ Fin))
9	8	imbi2d 230	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → (𝑦 × 𝐵) ∈ Fin)))
10		xpeq1 4678	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝐵) = (𝐴 × 𝐵))
11	10	eleq1d 2265	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 × 𝐵) ∈ Fin ↔ (𝐴 × 𝐵) ∈ Fin))
12	11	imbi2d 230	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → (𝐴 × 𝐵) ∈ Fin)))
13		0xp 4744	. . . . 5 ⊢ (∅ × 𝐵) = ∅
14		0fin 6954	. . . . 5 ⊢ ∅ ∈ Fin
15	13, 14	eqeltri 2269	. . . 4 ⊢ (∅ × 𝐵) ∈ Fin
16	15	a1i 9	. . 3 ⊢ (𝐵 ∈ Fin → (∅ × 𝐵) ∈ Fin)
17		xpeq1 4678	. . . . . . . 8 ⊢ (𝑦 = ∅ → (𝑦 × 𝐵) = (∅ × 𝐵))
18	17, 15	eqeltrdi 2287	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 × 𝐵) ∈ Fin)
19	18	a1i13 24	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝑦 = ∅ → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
20		sneq 3634	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
21	20	difeq2d 3282	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → (𝑦 ∖ {𝑧}) = (𝑦 ∖ {𝑤}))
22	21	xpeq1d 4687	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → ((𝑦 ∖ {𝑧}) × 𝐵) = ((𝑦 ∖ {𝑤}) × 𝐵))
23	22	eleq1d 2265	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin ↔ ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin))
24	23	imbi2d 230	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin)))
25	24	rspcv 2864	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin)))
26	25	adantl 277	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin)))
27		pm2.27 40	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → ((𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin) → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin))
28	27	ad2antlr 489	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → ((𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin) → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin))
29		vex 2766	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
30	29	snex 4219	. . . . . . . . . . . . . 14 ⊢ {𝑤} ∈ V
31		xpexg 4778	. . . . . . . . . . . . . 14 ⊢ (({𝑤} ∈ V ∧ 𝐵 ∈ Fin) → ({𝑤} × 𝐵) ∈ V)
32	30, 31	mpan 424	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Fin → ({𝑤} × 𝐵) ∈ V)
33		id 19	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Fin → 𝐵 ∈ Fin)
34		2ndconst 6289	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ V → (2nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto→𝐵)
35	29, 34	mp1i 10	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Fin → (2nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto→𝐵)
36		f1oen2g 6823	. . . . . . . . . . . . 13 ⊢ ((({𝑤} × 𝐵) ∈ V ∧ 𝐵 ∈ Fin ∧ (2nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto→𝐵) → ({𝑤} × 𝐵) ≈ 𝐵)
37	32, 33, 35, 36	syl3anc 1249	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Fin → ({𝑤} × 𝐵) ≈ 𝐵)
38		enfii 6944	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Fin ∧ ({𝑤} × 𝐵) ≈ 𝐵) → ({𝑤} × 𝐵) ∈ Fin)
39	37, 38	mpdan 421	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → ({𝑤} × 𝐵) ∈ Fin)
40	39	ad2antlr 489	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → ({𝑤} × 𝐵) ∈ Fin)
41		incom 3356	. . . . . . . . . . . . . 14 ⊢ ({𝑤} ∩ (𝑦 ∖ {𝑤})) = ((𝑦 ∖ {𝑤}) ∩ {𝑤})
42		disjdif 3524	. . . . . . . . . . . . . 14 ⊢ ({𝑤} ∩ (𝑦 ∖ {𝑤})) = ∅
43	41, 42	eqtr3i 2219	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∖ {𝑤}) ∩ {𝑤}) = ∅
44		xpdisj1 5095	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∖ {𝑤}) ∩ {𝑤}) = ∅ → (((𝑦 ∖ {𝑤}) × 𝐵) ∩ ({𝑤} × 𝐵)) = ∅)
45	43, 44	ax-mp 5	. . . . . . . . . . . 12 ⊢ (((𝑦 ∖ {𝑤}) × 𝐵) ∩ ({𝑤} × 𝐵)) = ∅
46		unfidisj 6992	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin ∧ ({𝑤} × 𝐵) ∈ Fin ∧ (((𝑦 ∖ {𝑤}) × 𝐵) ∩ ({𝑤} × 𝐵)) = ∅) → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin)
47	45, 46	mp3an3 1337	. . . . . . . . . . 11 ⊢ ((((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin ∧ ({𝑤} × 𝐵) ∈ Fin) → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin)
48		xpundir 4721	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∖ {𝑤}) ∪ {𝑤}) × 𝐵) = (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵))
49		fidifsnid 6941	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ Fin ∧ 𝑤 ∈ 𝑦) → ((𝑦 ∖ {𝑤}) ∪ {𝑤}) = 𝑦)
50	49	adantlr 477	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → ((𝑦 ∖ {𝑤}) ∪ {𝑤}) = 𝑦)
51	50	xpeq1d 4687	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → (((𝑦 ∖ {𝑤}) ∪ {𝑤}) × 𝐵) = (𝑦 × 𝐵))
52	48, 51	eqtr3id 2243	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) = (𝑦 × 𝐵))
53	52	eleq1d 2265	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → ((((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin ↔ (𝑦 × 𝐵) ∈ Fin))
54	47, 53	imbitrid 154	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → ((((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin ∧ ({𝑤} × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin))
55	40, 54	mpan2d 428	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → (((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin → (𝑦 × 𝐵) ∈ Fin))
56	26, 28, 55	3syld 57	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin))
57	56	ex 115	. . . . . . 7 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝑤 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
58	57	exlimdv 1833	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (∃𝑤 𝑤 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
59		fin0or 6956	. . . . . . 7 ⊢ (𝑦 ∈ Fin → (𝑦 = ∅ ∨ ∃𝑤 𝑤 ∈ 𝑦))
60	59	adantr 276	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝑦 = ∅ ∨ ∃𝑤 𝑤 ∈ 𝑦))
61	19, 58, 60	mpjaod 719	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin))
62	61	ex 115	. . . 4 ⊢ (𝑦 ∈ Fin → (𝐵 ∈ Fin → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
63	62	com23 78	. . 3 ⊢ (𝑦 ∈ Fin → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝐵 ∈ Fin → (𝑦 × 𝐵) ∈ Fin)))
64	3, 6, 9, 12, 16, 63	findcard 6958	. 2 ⊢ (𝐴 ∈ Fin → (𝐵 ∈ Fin → (𝐴 × 𝐵) ∈ Fin))
65	64	imp 124	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  Vcvv 2763   ∖ cdif 3154   ∪ cun 3155   ∩ cin 3156  ∅c0 3451  {csn 3623   class class class wbr 4034   × cxp 4662   ↾ cres 4666  –1-1-onto→wf1o 5258  2^nd c2nd 6206   ≈ cen 6806  Fincfn 6808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-1st 6207  df-2nd 6208  df-1o 6483  df-er 6601  df-en 6809  df-fin 6811

This theorem is referenced by:  3xpfi  7003  opabfi  7008  hashxp  10935  fsum2dlemstep  11616  fisumcom2  11620  fprod2dlemstep  11804  fprodcom2fi  11808  crth  12417  phimullem  12418  fsumdvdsmul  15311  lgsquadlem2  15403