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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.
StepHypRef Expression
1 xpeq1 4634 . . . . 5 (𝑥 = ∅ → (𝑥 × 𝐵) = (∅ × 𝐵))
21eleq1d 2244 . . . 4 (𝑥 = ∅ → ((𝑥 × 𝐵) ∈ Fin ↔ (∅ × 𝐵) ∈ Fin))
32imbi2d 230 . . 3 (𝑥 = ∅ → ((𝐵 ∈ Fin → (𝑥 × 𝐵) ∈ Fin) ↔ (𝐵 ∈ Fin → (∅ × 𝐵) ∈ Fin)))
4 xpeq1 4634 . . . . 5 (𝑥 = (𝑦 ∖ {𝑧}) → (𝑥 × 𝐵) = ((𝑦 ∖ {𝑧}) × 𝐵))
54eleq1d 2244 . . . 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 4634 . . . . 5 (𝑥 = 𝑦 → (𝑥 × 𝐵) = (𝑦 × 𝐵))
87eleq1d 2244 . . . 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 4634 . . . . 5 (𝑥 = 𝐴 → (𝑥 × 𝐵) = (𝐴 × 𝐵))
1110eleq1d 2244 . . . 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 4700 . . . . 5 (∅ × 𝐵) = ∅
14 0fin 6874 . . . . 5 ∅ ∈ Fin
1513, 14eqeltri 2248 . . . 4 (∅ × 𝐵) ∈ Fin
1615a1i 9 . . 3 (𝐵 ∈ Fin → (∅ × 𝐵) ∈ Fin)
17 xpeq1 4634 . . . . . . . 8 (𝑦 = ∅ → (𝑦 × 𝐵) = (∅ × 𝐵))
1817, 15eqeltrdi 2266 . . . . . . 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 3600 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → {𝑧} = {𝑤})
2120difeq2d 3251 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → (𝑦 ∖ {𝑧}) = (𝑦 ∖ {𝑤}))
2221xpeq1d 4643 . . . . . . . . . . . . 13 (𝑧 = 𝑤 → ((𝑦 ∖ {𝑧}) × 𝐵) = ((𝑦 ∖ {𝑤}) × 𝐵))
2322eleq1d 2244 . . . . . . . . . . . 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 2835 . . . . . . . . . 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 2738 . . . . . . . . . . . . . . 15 𝑤 ∈ V
3029snex 4180 . . . . . . . . . . . . . 14 {𝑤} ∈ V
31 xpexg 4734 . . . . . . . . . . . . . 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 6213 . . . . . . . . . . . . . 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 6745 . . . . . . . . . . . . 13 ((({𝑤} × 𝐵) ∈ V ∧ 𝐵 ∈ Fin ∧ (2nd ↾ ({𝑤} × 𝐵)):({𝑤} × 𝐵)–1-1-onto𝐵) → ({𝑤} × 𝐵) ≈ 𝐵)
3732, 33, 35, 36syl3anc 1238 . . . . . . . . . . . 12 (𝐵 ∈ Fin → ({𝑤} × 𝐵) ≈ 𝐵)
38 enfii 6864 . . . . . . . . . . . 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 3325 . . . . . . . . . . . . . 14 ({𝑤} ∩ (𝑦 ∖ {𝑤})) = ((𝑦 ∖ {𝑤}) ∩ {𝑤})
42 disjdif 3493 . . . . . . . . . . . . . 14 ({𝑤} ∩ (𝑦 ∖ {𝑤})) = ∅
4341, 42eqtr3i 2198 . . . . . . . . . . . . 13 ((𝑦 ∖ {𝑤}) ∩ {𝑤}) = ∅
44 xpdisj1 5045 . . . . . . . . . . . . 13 (((𝑦 ∖ {𝑤}) ∩ {𝑤}) = ∅ → (((𝑦 ∖ {𝑤}) × 𝐵) ∩ ({𝑤} × 𝐵)) = ∅)
4543, 44ax-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)
4745, 46mp3an3 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 ∧ 𝑤𝑦) → ((𝑦 ∖ {𝑤}) ∪ {𝑤}) = 𝑦)
5049adantlr 477 . . . . . . . . . . . . . 14 (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤𝑦) → ((𝑦 ∖ {𝑤}) ∪ {𝑤}) = 𝑦)
5150xpeq1d 4643 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤𝑦) → (((𝑦 ∖ {𝑤}) ∪ {𝑤}) × 𝐵) = (𝑦 × 𝐵))
5248, 51eqtr3id 2222 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤𝑦) → (((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) = (𝑦 × 𝐵))
5352eleq1d 2244 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑤𝑦) → ((((𝑦 ∖ {𝑤}) × 𝐵) ∪ ({𝑤} × 𝐵)) ∈ Fin ↔ (𝑦 × 𝐵) ∈ Fin))
5447, 53syl5ib 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 1817 . . . . . 6 ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (∃𝑤 𝑤𝑦 → (∀𝑧𝑦 (𝐵 ∈ Fin → ((𝑦 ∖ {𝑧}) × 𝐵) ∈ Fin) → (𝑦 × 𝐵) ∈ Fin)))
59 fin0or 6876 . . . . . . 7 (𝑦 ∈ Fin → (𝑦 = ∅ ∨ ∃𝑤 𝑤𝑦))
6059adantr 276 . . . . . 6 ((𝑦 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝑦 = ∅ ∨ ∃𝑤 𝑤𝑦))
6119, 58, 60mpjaod 718 . . . . 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 6878 . 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 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-ontowf1o 5207  2nd 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
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