xpfi - Intuitionistic Logic Explorer

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Theorem xpfi 6919

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 4634	. . . . 5 ⊢ (𝑥 = ∅ → (𝑥 × 𝐵) = (∅ × 𝐵))
2	1	eleq1d 2244	. . . 4 ⊢ (𝑥 = ∅ → ((𝑥 × 𝐵) ∈ Fin ↔ (∅ × 𝐵) ∈ Fin))
3	2	imbi2d 230	. . 3 ⊢ (𝑥 = ∅ → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → (∅ × 𝐵) ∈ Fin)))
4		xpeq1 4634	. . . . 5 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝑥 × 𝐵) = ((𝑦 ∖ {𝑧}) × 𝐵))
5	4	eleq1d 2244	. . . 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 4634	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 × 𝐵) = (𝑦 × 𝐵))
8	7	eleq1d 2244	. . . 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 4634	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝐵) = (𝐴 × 𝐵))
11	10	eleq1d 2244	. . . 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 4700	. . . . 5 ⊢ (∅ × 𝐵) = ∅
14		0fin 6874	. . . . 5 ⊢ ∅ ∈ Fin
15	13, 14	eqeltri 2248	. . . 4 ⊢ (∅ × 𝐵) ∈ Fin
16	15	a1i 9	. . 3 ⊢ (𝐵 ∈ Fin → (∅ × 𝐵) ∈ Fin)
17		xpeq1 4634	. . . . . . . 8 ⊢ (𝑦 = ∅ → (𝑦 × 𝐵) = (∅ × 𝐵))
18	17, 15	eqeltrdi 2266	. . . . . . 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 3600	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → {𝑧} = {𝑤})
21	20	difeq2d 3251	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → (𝑦 ∖ {𝑧}) = (𝑦 ∖ {𝑤}))
22	21	xpeq1d 4643	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → ((𝑦 ∖ {𝑧}) × 𝐵) = ((𝑦 ∖ {𝑤}) × 𝐵))
23	22	eleq1d 2244	. . . . . . . . . . . 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 2835	. . . . . . . . . 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 2738	. . . . . . . . . . . . . . 15 ⊢ 𝑤 ∈ V
30	29	snex 4180	. . . . . . . . . . . . . 14 ⊢ {𝑤} ∈ V
31		xpexg 4734	. . . . . . . . . . . . . 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 6213	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ V → (2^nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto→𝐵)
35	29, 34	mp1i 10	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Fin → (2^nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto→𝐵)
36		f1oen2g 6745	. . . . . . . . . . . . 13 ⊢ ((({𝑤} × 𝐵) ∈ V ∧ 𝐵 ∈ Fin ∧ (2^nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto→𝐵) → ({𝑤} × 𝐵) ≈ 𝐵)
37	32, 33, 35, 36	syl3anc 1238	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Fin → ({𝑤} × 𝐵) ≈ 𝐵)
38		enfii 6864	. . . . . . . . . . . 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 3325	. . . . . . . . . . . . . 14 ⊢ ({𝑤} ∩ (𝑦 ∖ {𝑤})) = ((𝑦 ∖ {𝑤}) ∩ {𝑤})
42		disjdif 3493	. . . . . . . . . . . . . 14 ⊢ ({𝑤} ∩ (𝑦 ∖ {𝑤})) = ∅
43	41, 42	eqtr3i 2198	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∖ {𝑤}) ∩ {𝑤}) = ∅
44		xpdisj1 5045	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∖ {𝑤}) ∩ {𝑤}) = ∅ → (((𝑦 ∖ {𝑤}) × 𝐵) ∩ ({𝑤} × 𝐵)) = ∅)
45	43, 44	ax-mp 5	. . . . . . . . . . . 12 ⊢ (((𝑦 ∖ {𝑤}) × 𝐵) ∩ ({𝑤} × 𝐵)) = ∅
46		unfidisj 6911	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin ∧ ({𝑤} × 𝐵) ∈ Fin ∧ (((𝑦 ∖ {𝑤}) × 𝐵) ∩ ({𝑤} × 𝐵)) = ∅) → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin)
47	45, 46	mp3an3 1326	. . . . . . . . . . 11 ⊢ ((((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin ∧ ({𝑤} × 𝐵) ∈ Fin) → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin)
48		xpundir 4677	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∖ {𝑤}) ∪ {𝑤}) × 𝐵) = (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵))
49		fidifsnid 6861	. . . . . . . . . . . . . . 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 4643	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → (((𝑦 ∖ {𝑤}) ∪ {𝑤}) × 𝐵) = (𝑦 × 𝐵))
52	48, 51	eqtr3id 2222	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) = (𝑦 × 𝐵))
53	52	eleq1d 2244	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤 ∈ 𝑦) → ((((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin ↔ (𝑦 × 𝐵) ∈ Fin))
54	47, 53	syl5ib 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 1817	. . . . . 6 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (∃𝑤 𝑤 ∈ 𝑦 → (∀𝑧 ∈ 𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
59		fin0or 6876	. . . . . . 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 718	. . . . 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 6878	. 2 ⊢ (𝐴 ∈ Fin → (𝐵 ∈ Fin → (𝐴 × 𝐵) ∈ Fin))
65	64	imp 124	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∨ wo 708 = wceq 1353 ∃wex 1490 ∈ wcel 2146 ∀wral 2453 Vcvv 2735 ∖ cdif 3124 ∪ cun 3125 ∩ cin 3126 ∅c0 3420 {csn 3589 class class class wbr 3998 × cxp 4618 ↾ cres 4622 –1-1-onto→wf1o 5207 2^nd c2nd 6130 ≈ cen 6728 Fincfn 6730

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 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-iord 4360 df-on 4362 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-1st 6131 df-2nd 6132 df-1o 6407 df-er 6525 df-en 6731 df-fin 6733

This theorem is referenced by: 3xpfi 6920 hashxp 10772 fsum2dlemstep 11408 fisumcom2 11412 fprod2dlemstep 11596 fprodcom2fi 11600 crth 12189 phimullem 12190

W3C validator