Theorem xpfi 7105

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 4733	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 × 𝐵) = (∅ × 𝐵))
2	1	eleq1d 2298	. . . 4 ⊢ (𝑥 = ∅ → ((𝑥 × 𝐵) ∈ Fin ↔ (∅ × 𝐵) ∈ Fin))
3	2	imbi2d 230	. . 3 ⊢ (𝑥 = ∅ → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → (∅ × 𝐵) ∈ Fin)))
4		xpeq1 4733	. . . . 5 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝑥 × 𝐵) = ((𝑦 ∖ {𝑧}) × 𝐵))
5	4	eleq1d 2298	. . . 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 4733	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 × 𝐵) = (𝑦 × 𝐵))
8	7	eleq1d 2298	. . . 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 4733	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝐵) = (𝐴 × 𝐵))
11	10	eleq1d 2298	. . . 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 4799	. . . . 5 ⊢ (∅ × 𝐵) = ∅
14		0fi 7054	. . . . 5 ⊢ ∅ ∈ Fin
15	13, 14	eqeltri 2302	. . . 4 ⊢ (∅ × 𝐵) ∈ Fin
16	15	a1i 9	. . 3 ⊢ (𝐵 ∈ Fin → (∅ × 𝐵) ∈ Fin)
17		xpeq1 4733	. . . . . . . 8 ⊢ (𝑦 = ∅ → (𝑦 × 𝐵) = (∅ × 𝐵))
18	17, 15	eqeltrdi 2320	. . . . . . 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 3677	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
21	20	difeq2d 3322	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → (𝑦 ∖ {𝑧}) = (𝑦 ∖ {𝑤}))
22	21	xpeq1d 4742	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → ((𝑦 ∖ {𝑧}) × 𝐵) = ((𝑦 ∖ {𝑤}) × 𝐵))
23	22	eleq1d 2298	. . . . . . . . . . . 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 2903	. . . . . . . . . 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 2802	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
30	29	snex 4269	. . . . . . . . . . . . . 14 ⊢ {𝑤} ∈ V
31		xpexg 4833	. . . . . . . . . . . . . 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 6374	. . . . . . . . . . . . . 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 6914	. . . . . . . . . . . . 13 ⊢ ((({𝑤} × 𝐵) ∈ V ∧ 𝐵 ∈ Fin ∧ (2nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto→𝐵) → ({𝑤} × 𝐵) ≈ 𝐵)
37	32, 33, 35, 36	syl3anc 1271	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Fin → ({𝑤} × 𝐵) ≈ 𝐵)
38		enfii 7044	. . . . . . . . . . . 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 3396	. . . . . . . . . . . . . 14 ⊢ ({𝑤} ∩ (𝑦 ∖ {𝑤})) = ((𝑦 ∖ {𝑤}) ∩ {𝑤})
42		disjdif 3564	. . . . . . . . . . . . . 14 ⊢ ({𝑤} ∩ (𝑦 ∖ {𝑤})) = ∅
43	41, 42	eqtr3i 2252	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∖ {𝑤}) ∩ {𝑤}) = ∅
44		xpdisj1 5153	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∖ {𝑤}) ∩ {𝑤}) = ∅ → (((𝑦 ∖ {𝑤}) × 𝐵) ∩ ({𝑤} × 𝐵)) = ∅)
45	43, 44	ax-mp 5	. . . . . . . . . . . 12 ⊢ (((𝑦 ∖ {𝑤}) × 𝐵) ∩ ({𝑤} × 𝐵)) = ∅
46		unfidisj 7095	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin ∧ ({𝑤} × 𝐵) ∈ Fin ∧ (((𝑦 ∖ {𝑤}) × 𝐵) ∩ ({𝑤} × 𝐵)) = ∅) → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin)
47	45, 46	mp3an3 1360	. . . . . . . . . . 11 ⊢ ((((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin ∧ ({𝑤} × 𝐵) ∈ Fin) → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin)
48		xpundir 4776	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∖ {𝑤}) ∪ {𝑤}) × 𝐵) = (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵))
49		fidifsnid 7041	. . . . . . . . . . . . . . 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 4742	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → (((𝑦 ∖ {𝑤}) ∪ {𝑤}) × 𝐵) = (𝑦 × 𝐵))
52	48, 51	eqtr3id 2276	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) = (𝑦 × 𝐵))
53	52	eleq1d 2298	. . . . . . . . . . 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 1865	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (∃𝑤 𝑤 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
59		fin0or 7056	. . . . . . 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 723	. . . . 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 7058	. 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 713   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  Vcvv 2799   ∖ cdif 3194   ∪ cun 3195   ∩ cin 3196  ∅c0 3491  {csn 3666   class class class wbr 4083   × cxp 4717   ↾ cres 4721  –1-1-onto→wf1o 5317  2^nd c2nd 6291   ≈ cen 6893  Fincfn 6895

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1st 6292  df-2nd 6293  df-1o 6568  df-er 6688  df-en 6896  df-fin 6898

This theorem is referenced by:  3xpfi  7106  opabfi  7111  hashxp  11061  fsum2dlemstep  11960  fisumcom2  11964  fprod2dlemstep  12148  fprodcom2fi  12152  crth  12761  phimullem  12762  fsumdvdsmul  15680  lgsquadlem2  15772