Theorem xpfi 9046

Description: The Cartesian product of two finite sets is finite. (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 5602	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 × 𝐵) = (∅ × 𝐵))
2	1	eleq1d 2824	. . . 4 ⊢ (𝑥 = ∅ → ((𝑥 × 𝐵) ∈ Fin ↔ (∅ × 𝐵) ∈ Fin))
3	2	imbi2d 340	. . 3 ⊢ (𝑥 = ∅ → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → (∅ × 𝐵) ∈ Fin)))
4		xpeq1 5602	. . . . 5 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝑥 × 𝐵) = ((𝑦 ∖ {𝑧}) × 𝐵))
5	4	eleq1d 2824	. . . 4 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → ((𝑥 × 𝐵) ∈ Fin ↔ ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin))
6	5	imbi2d 340	. . 3 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin)))
7		xpeq1 5602	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 × 𝐵) = (𝑦 × 𝐵))
8	7	eleq1d 2824	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 × 𝐵) ∈ Fin ↔ (𝑦 × 𝐵) ∈ Fin))
9	8	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → (𝑦 × 𝐵) ∈ Fin)))
10		xpeq1 5602	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝐵) = (𝐴 × 𝐵))
11	10	eleq1d 2824	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 × 𝐵) ∈ Fin ↔ (𝐴 × 𝐵) ∈ Fin))
12	11	imbi2d 340	. . 3 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → (𝐴 × 𝐵) ∈ Fin)))
13		0xp 5683	. . . . 5 ⊢ (∅ × 𝐵) = ∅
14		0fin 8919	. . . . 5 ⊢ ∅ ∈ Fin
15	13, 14	eqeltri 2836	. . . 4 ⊢ (∅ × 𝐵) ∈ Fin
16	15	a1i 11	. . 3 ⊢ (𝐵 ∈ Fin → (∅ × 𝐵) ∈ Fin)
17		neq0 4284	. . . . . . 7 ⊢ (¬ 𝑦 = ∅ ↔ ∃𝑤 𝑤 ∈ 𝑦)
18		sneq 4576	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
19	18	difeq2d 4061	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑦 ∖ {𝑧}) = (𝑦 ∖ {𝑤}))
20	19	xpeq1d 5617	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → ((𝑦 ∖ {𝑧}) × 𝐵) = ((𝑦 ∖ {𝑤}) × 𝐵))
21	20	eleq1d 2824	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin ↔ ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin))
22	21	imbi2d 340	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → ((𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin)))
23	22	rspcv 3555	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin)))
24	23	adantl 481	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin)))
25		pm2.27 42	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → ((𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin) → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin))
26	25	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → ((𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin) → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin))
27		snex 5357	. . . . . . . . . . . . . . 15 ⊢ {𝑤} ∈ V
28		xpexg 7591	. . . . . . . . . . . . . . 15 ⊢ (({𝑤} ∈ V ∧ 𝐵 ∈ Fin) → ({𝑤} × 𝐵) ∈ V)
29	27, 28	mpan 686	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ Fin → ({𝑤} × 𝐵) ∈ V)
30		id 22	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ Fin → 𝐵 ∈ Fin)
31		vex 3434	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
32		2ndconst 7925	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ V → (2nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto→𝐵)
33	31, 32	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ Fin → (2nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto→𝐵)
34		f1oen2g 8727	. . . . . . . . . . . . . 14 ⊢ ((({𝑤} × 𝐵) ∈ V ∧ 𝐵 ∈ Fin ∧ (2nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto→𝐵) → ({𝑤} × 𝐵) ≈ 𝐵)
35	29, 30, 33, 34	syl3anc 1369	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Fin → ({𝑤} × 𝐵) ≈ 𝐵)
36		enfii 8937	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ Fin ∧ ({𝑤} × 𝐵) ≈ 𝐵) → ({𝑤} × 𝐵) ∈ Fin)
37	35, 36	mpdan 683	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Fin → ({𝑤} × 𝐵) ∈ Fin)
38	37	ad2antlr 723	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → ({𝑤} × 𝐵) ∈ Fin)
39		unfi 8920	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin ∧ ({𝑤} × 𝐵) ∈ Fin) → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin)
40		xpundir 5655	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∖ {𝑤}) ∪ {𝑤}) × 𝐵) = (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵))
41		difsnid 4748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ 𝑦 → ((𝑦 ∖ {𝑤}) ∪ {𝑤}) = 𝑦)
42	41	xpeq1d 5617	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ 𝑦 → (((𝑦 ∖ {𝑤}) ∪ {𝑤}) × 𝐵) = (𝑦 × 𝐵))
43	40, 42	eqtr3id 2793	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ 𝑦 → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) = (𝑦 × 𝐵))
44	43	eleq1d 2824	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝑦 → ((((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin ↔ (𝑦 × 𝐵) ∈ Fin))
45	44	biimpd 228	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ 𝑦 → ((((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin → (𝑦 × 𝐵) ∈ Fin))
46	45	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → ((((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin → (𝑦 × 𝐵) ∈ Fin))
47	39, 46	syl5 34	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → ((((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin ∧ ({𝑤} × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin))
48	38, 47	mpan2d 690	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → (((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin → (𝑦 × 𝐵) ∈ Fin))
49	24, 26, 48	3syld 60	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin))
50	49	ex 412	. . . . . . . 8 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝑤 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
51	50	exlimdv 1939	. . . . . . 7 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (∃𝑤 𝑤 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
52	17, 51	syl5bi 241	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (¬ 𝑦 = ∅ → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
53		xpeq1 5602	. . . . . . . 8 ⊢ (𝑦 = ∅ → (𝑦 × 𝐵) = (∅ × 𝐵))
54	53, 15	eqeltrdi 2848	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 × 𝐵) ∈ Fin)
55	54	a1d 25	. . . . . 6 ⊢ (𝑦 = ∅ → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin))
56	52, 55	pm2.61d2 181	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin))
57	56	ex 412	. . . 4 ⊢ (𝑦 ∈ Fin → (𝐵 ∈ Fin → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
58	57	com23 86	. . 3 ⊢ (𝑦 ∈ Fin → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝐵 ∈ Fin → (𝑦 × 𝐵) ∈ Fin)))
59	3, 6, 9, 12, 16, 58	findcard 8911	. 2 ⊢ (𝐴 ∈ Fin → (𝐵 ∈ Fin → (𝐴 × 𝐵) ∈ Fin))
60	59	imp 406	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1785   ∈ wcel 2109  ∀wral 3065  Vcvv 3430   ∖ cdif 3888   ∪ cun 3889  ∅c0 4261  {csn 4566   class class class wbr 5078   × cxp 5586   ↾ cres 5590  –1-1-onto→wf1o 6429  2^nd c2nd 7816   ≈ cen 8704  Fincfn 8707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-om 7701  df-1st 7817  df-2nd 7818  df-1o 8281  df-en 8708  df-fin 8711

This theorem is referenced by:  3xpfi  9047  mapfi  9076  fsuppxpfi  9106  infxpenlem  9753  ficardadju  9939  ackbij1lem9  9968  ackbij1lem10  9969  hashxplem  14129  hashmap  14131  fsum2dlem  15463  fsumcom2  15467  ackbijnn  15521  fprod2dlem  15671  fprodcom2  15675  rexpen  15918  crth  16460  phimullem  16461  prmreclem3  16600  gsumcom3fi  19561  ablfaclem3  19671  gsumdixp  19829  frlmbas3  20964  gsumbagdiagOLD  21123  psrass1lemOLD  21124  gsumbagdiag  21126  psrass1lem  21127  evlslem2  21270  mamudm  21518  mamufacex  21519  mamures  21520  mamucl  21529  mamudi  21531  mamudir  21532  mamuvs1  21533  mamuvs2  21534  matsca2  21550  matbas2  21551  matplusg2  21557  matvsca2  21558  matplusgcell  21563  matsubgcell  21564  matvscacell  21566  matgsum  21567  mamumat1cl  21569  mattposcl  21583  mdetrsca  21733  mdetunilem9  21750  pmatcoe1fsupp  21831  tsmsxplem1  23285  tsmsxplem2  23286  tsmsxp  23287  i1fadd  24840  i1fmul  24841  itg1addlem4  24844  itg1addlem4OLD  24845  fsumdvdsmul  26325  fsumvma  26342  lgsquadlem1  26509  lgsquadlem2  26510  lgsquadlem3  26511  relfi  30920  fsumiunle  31122  matdim  31677  fedgmullem1  31689  sibfof  32286  hgt750lemb  32615  erdszelem10  33141  matunitlindflem2  35753  matunitlindf  35754  poimirlem26  35782  poimirlem27  35783  poimirlem28  35784  cntotbnd  35933  sticksstones22  40104  pellex  40637  mnringmulrcld  41799  fourierdlem42  43644  etransclem44  43773  etransclem45  43774  etransclem47  43776