ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpfi GIF version

Theorem xpfi 6959
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.
StepHypRef Expression
1 xpeq1 4658 . . . . 5 (𝑥 = ∅ → (𝑥 × 𝐵) = (∅ × 𝐵))
21eleq1d 2258 . . . 4 (𝑥 = ∅ → ((𝑥 × 𝐵) ∈ Fin ↔ (∅ × 𝐵) ∈ Fin))
32imbi2d 230 . . 3 (𝑥 = ∅ → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → (∅ × 𝐵) ∈ Fin)))
4 xpeq1 4658 . . . . 5 (𝑥 = (𝑦 ∖ {𝑧}) → (𝑥 × 𝐵) = ((𝑦 ∖ {𝑧}) × 𝐵))
54eleq1d 2258 . . . 4 (𝑥 = (𝑦 ∖ {𝑧}) → ((𝑥 × 𝐵) ∈ Fin ↔ ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin))
65imbi2d 230 . . 3 (𝑥 = (𝑦 ∖ {𝑧}) → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin)))
7 xpeq1 4658 . . . . 5 (𝑥 = 𝑦 → (𝑥 × 𝐵) = (𝑦 × 𝐵))
87eleq1d 2258 . . . 4 (𝑥 = 𝑦 → ((𝑥 × 𝐵) ∈ Fin ↔ (𝑦 × 𝐵) ∈ Fin))
98imbi2d 230 . . 3 (𝑥 = 𝑦 → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → (𝑦 × 𝐵) ∈ Fin)))
10 xpeq1 4658 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝐵) = (𝐴 × 𝐵))
1110eleq1d 2258 . . . 4 (𝑥 = 𝐴 → ((𝑥 × 𝐵) ∈ Fin ↔ (𝐴 × 𝐵) ∈ Fin))
1211imbi2d 230 . . 3 (𝑥 = 𝐴 → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → (𝐴 × 𝐵) ∈ Fin)))
13 0xp 4724 . . . . 5 (∅ × 𝐵) = ∅
14 0fin 6913 . . . . 5 ∅ ∈ Fin
1513, 14eqeltri 2262 . . . 4 (∅ × 𝐵) ∈ Fin
1615a1i 9 . . 3 (𝐵 ∈ Fin → (∅ × 𝐵) ∈ Fin)
17 xpeq1 4658 . . . . . . . 8 (𝑦 = ∅ → (𝑦 × 𝐵) = (∅ × 𝐵))
1817, 15eqeltrdi 2280 . . . . . . 7 (𝑦 = ∅ → (𝑦 × 𝐵) ∈ Fin)
1918a1i13 24 . . . . . 6 ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝑦 = ∅ → (∀𝑧𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
20 sneq 3618 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → {𝑧} = {𝑤})
2120difeq2d 3268 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → (𝑦 ∖ {𝑧}) = (𝑦 ∖ {𝑤}))
2221xpeq1d 4667 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → ((𝑦 ∖ {𝑧}) × 𝐵) = ((𝑦 ∖ {𝑤}) × 𝐵))
2322eleq1d 2258 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin ↔ ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin))
2423imbi2d 230 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin)))
2524rspcv 2852 . . . . . . . . . 10 (𝑤𝑦 → (∀𝑧𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin)))
2625adantl 277 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤𝑦) → (∀𝑧𝑦 (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))
2827ad2antlr 489 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤𝑦) → ((𝐵 ∈ Fin → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin) → ((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin))
29 vex 2755 . . . . . . . . . . . . . . 15 𝑤 ∈ V
3029snex 4203 . . . . . . . . . . . . . 14 {𝑤} ∈ V
31 xpexg 4758 . . . . . . . . . . . . . 14 (({𝑤} ∈ V ∧ 𝐵 ∈ Fin) → ({𝑤} × 𝐵) ∈ V)
3230, 31mpan 424 . . . . . . . . . . . . 13 (𝐵 ∈ Fin → ({𝑤} × 𝐵) ∈ V)
33 id 19 . . . . . . . . . . . . 13 (𝐵 ∈ Fin → 𝐵 ∈ Fin)
34 2ndconst 6248 . . . . . . . . . . . . . 14 (𝑤 ∈ V → (2nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto𝐵)
3529, 34mp1i 10 . . . . . . . . . . . . 13 (𝐵 ∈ Fin → (2nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto𝐵)
36 f1oen2g 6782 . . . . . . . . . . . . 13 ((({𝑤} × 𝐵) ∈ V ∧ 𝐵 ∈ Fin ∧ (2nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto𝐵) → ({𝑤} × 𝐵) ≈ 𝐵)
3732, 33, 35, 36syl3anc 1249 . . . . . . . . . . . 12 (𝐵 ∈ Fin → ({𝑤} × 𝐵) ≈ 𝐵)
38 enfii 6903 . . . . . . . . . . . 12 ((𝐵 ∈ Fin ∧ ({𝑤} × 𝐵) ≈ 𝐵) → ({𝑤} × 𝐵) ∈ Fin)
3937, 38mpdan 421 . . . . . . . . . . 11 (𝐵 ∈ Fin → ({𝑤} × 𝐵) ∈ Fin)
4039ad2antlr 489 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤𝑦) → ({𝑤} × 𝐵) ∈ Fin)
41 incom 3342 . . . . . . . . . . . . . 14 ({𝑤} ∩ (𝑦 ∖ {𝑤})) = ((𝑦 ∖ {𝑤}) ∩ {𝑤})
42 disjdif 3510 . . . . . . . . . . . . . 14 ({𝑤} ∩ (𝑦 ∖ {𝑤})) = ∅
4341, 42eqtr3i 2212 . . . . . . . . . . . . 13 ((𝑦 ∖ {𝑤}) ∩ {𝑤}) = ∅
44 xpdisj1 5071 . . . . . . . . . . . . 13 (((𝑦 ∖ {𝑤}) ∩ {𝑤}) = ∅ → (((𝑦 ∖ {𝑤}) × 𝐵) ∩ ({𝑤} × 𝐵)) = ∅)
4543, 44ax-mp 5 . . . . . . . . . . . 12 (((𝑦 ∖ {𝑤}) × 𝐵) ∩ ({𝑤} × 𝐵)) = ∅
46 unfidisj 6951 . . . . . . . . . . . 12 ((((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin ∧ ({𝑤} × 𝐵) ∈ Fin ∧ (((𝑦 ∖ {𝑤}) × 𝐵) ∩ ({𝑤} × 𝐵)) = ∅) → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin)
4745, 46mp3an3 1337 . . . . . . . . . . 11 ((((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin ∧ ({𝑤} × 𝐵) ∈ Fin) → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin)
48 xpundir 4701 . . . . . . . . . . . . 13 (((𝑦 ∖ {𝑤}) ∪ {𝑤}) × 𝐵) = (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵))
49 fidifsnid 6900 . . . . . . . . . . . . . . 15 ((𝑦 ∈ Fin ∧ 𝑤𝑦) → ((𝑦 ∖ {𝑤}) ∪ {𝑤}) = 𝑦)
5049adantlr 477 . . . . . . . . . . . . . 14 (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤𝑦) → ((𝑦 ∖ {𝑤}) ∪ {𝑤}) = 𝑦)
5150xpeq1d 4667 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤𝑦) → (((𝑦 ∖ {𝑤}) ∪ {𝑤}) × 𝐵) = (𝑦 × 𝐵))
5248, 51eqtr3id 2236 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤𝑦) → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) = (𝑦 × 𝐵))
5352eleq1d 2258 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤𝑦) → ((((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin ↔ (𝑦 × 𝐵) ∈ Fin))
5447, 53imbitrid 154 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤𝑦) → ((((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin ∧ ({𝑤} × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin))
5540, 54mpan2d 428 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤𝑦) → (((𝑦 ∖ {𝑤}) × 𝐵) ∈ Fin → (𝑦 × 𝐵) ∈ Fin))
5626, 28, 553syld 57 . . . . . . . 8 (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤𝑦) → (∀𝑧𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin))
5756ex 115 . . . . . . 7 ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝑤𝑦 → (∀𝑧𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
5857exlimdv 1830 . . . . . 6 ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (∃𝑤 𝑤𝑦 → (∀𝑧𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
59 fin0or 6915 . . . . . . 7 (𝑦 ∈ Fin → (𝑦 = ∅ ∨ ∃𝑤 𝑤𝑦))
6059adantr 276 . . . . . 6 ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝑦 = ∅ ∨ ∃𝑤 𝑤𝑦))
6119, 58, 60mpjaod 719 . . . . 5 ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (∀𝑧𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin))
6261ex 115 . . . 4 (𝑦 ∈ Fin → (𝐵 ∈ Fin → (∀𝑧𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
6362com23 78 . . 3 (𝑦 ∈ Fin → (∀𝑧𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝐵 ∈ Fin → (𝑦 × 𝐵) ∈ Fin)))
643, 6, 9, 12, 16, 63findcard 6917 . 2 (𝐴 ∈ Fin → (𝐵 ∈ Fin → (𝐴 × 𝐵) ∈ Fin))
6564imp 124 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 × 𝐵) ∈ Fin)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 709   = wceq 1364  wex 1503  wcel 2160  wral 2468  Vcvv 2752  cdif 3141  cun 3142  cin 3143  c0 3437  {csn 3607   class class class wbr 4018   × cxp 4642  cres 4646  1-1-ontowf1o 5234  2nd c2nd 6165  cen 6765  Fincfn 6767
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6166  df-2nd 6167  df-1o 6442  df-er 6560  df-en 6768  df-fin 6770
This theorem is referenced by:  3xpfi  6960  hashxp  10841  fsum2dlemstep  11477  fisumcom2  11481  fprod2dlemstep  11665  fprodcom2fi  11669  crth  12259  phimullem  12260
  Copyright terms: Public domain W3C validator