MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgrest Structured version   Visualization version   GIF version

Theorem tgrest 22645
Description: A subspace can be generated by restricted sets from a basis for the original topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
tgrest ((𝐵𝑉𝐴𝑊) → (topGen‘(𝐵t 𝐴)) = ((topGen‘𝐵) ↾t 𝐴))

Proof of Theorem tgrest
Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 7437 . . . . 5 (𝐵t 𝐴) ∈ V
2 eltg3 22447 . . . . 5 ((𝐵t 𝐴) ∈ V → (𝑥 ∈ (topGen‘(𝐵t 𝐴)) ↔ ∃𝑦(𝑦 ⊆ (𝐵t 𝐴) ∧ 𝑥 = 𝑦)))
31, 2ax-mp 5 . . . 4 (𝑥 ∈ (topGen‘(𝐵t 𝐴)) ↔ ∃𝑦(𝑦 ⊆ (𝐵t 𝐴) ∧ 𝑥 = 𝑦))
4 simpll 766 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → 𝐵𝑉)
5 funmpt 6583 . . . . . . . . . 10 Fun (𝑥𝐵 ↦ (𝑥𝐴))
65a1i 11 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → Fun (𝑥𝐵 ↦ (𝑥𝐴)))
7 restval 17368 . . . . . . . . . . . 12 ((𝐵𝑉𝐴𝑊) → (𝐵t 𝐴) = ran (𝑥𝐵 ↦ (𝑥𝐴)))
87sseq2d 4013 . . . . . . . . . . 11 ((𝐵𝑉𝐴𝑊) → (𝑦 ⊆ (𝐵t 𝐴) ↔ 𝑦 ⊆ ran (𝑥𝐵 ↦ (𝑥𝐴))))
98biimpa 478 . . . . . . . . . 10 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → 𝑦 ⊆ ran (𝑥𝐵 ↦ (𝑥𝐴)))
10 vex 3479 . . . . . . . . . . . . 13 𝑥 ∈ V
1110inex1 5316 . . . . . . . . . . . 12 (𝑥𝐴) ∈ V
1211rgenw 3066 . . . . . . . . . . 11 𝑥𝐵 (𝑥𝐴) ∈ V
13 eqid 2733 . . . . . . . . . . . 12 (𝑥𝐵 ↦ (𝑥𝐴)) = (𝑥𝐵 ↦ (𝑥𝐴))
1413fnmpt 6687 . . . . . . . . . . 11 (∀𝑥𝐵 (𝑥𝐴) ∈ V → (𝑥𝐵 ↦ (𝑥𝐴)) Fn 𝐵)
15 fnima 6677 . . . . . . . . . . 11 ((𝑥𝐵 ↦ (𝑥𝐴)) Fn 𝐵 → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵) = ran (𝑥𝐵 ↦ (𝑥𝐴)))
1612, 14, 15mp2b 10 . . . . . . . . . 10 ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵) = ran (𝑥𝐵 ↦ (𝑥𝐴))
179, 16sseqtrrdi 4032 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → 𝑦 ⊆ ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵))
18 ssimaexg 6973 . . . . . . . . 9 ((𝐵𝑉 ∧ Fun (𝑥𝐵 ↦ (𝑥𝐴)) ∧ 𝑦 ⊆ ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵)) → ∃𝑧(𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)))
194, 6, 17, 18syl3anc 1372 . . . . . . . 8 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → ∃𝑧(𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)))
20 df-ima 5688 . . . . . . . . . . . . . . . . 17 ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ran ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧)
21 resmpt 6035 . . . . . . . . . . . . . . . . . . 19 (𝑧𝐵 → ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧) = (𝑥𝑧 ↦ (𝑥𝐴)))
2221adantl 483 . . . . . . . . . . . . . . . . . 18 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧) = (𝑥𝑧 ↦ (𝑥𝐴)))
2322rneqd 5935 . . . . . . . . . . . . . . . . 17 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ran ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧) = ran (𝑥𝑧 ↦ (𝑥𝐴)))
2420, 23eqtrid 2785 . . . . . . . . . . . . . . . 16 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ran (𝑥𝑧 ↦ (𝑥𝐴)))
2524unieqd 4921 . . . . . . . . . . . . . . 15 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ran (𝑥𝑧 ↦ (𝑥𝐴)))
2611dfiun3 5963 . . . . . . . . . . . . . . 15 𝑥𝑧 (𝑥𝐴) = ran (𝑥𝑧 ↦ (𝑥𝐴))
2725, 26eqtr4di 2791 . . . . . . . . . . . . . 14 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = 𝑥𝑧 (𝑥𝐴))
28 iunin1 5074 . . . . . . . . . . . . . 14 𝑥𝑧 (𝑥𝐴) = ( 𝑥𝑧 𝑥𝐴)
2927, 28eqtrdi 2789 . . . . . . . . . . . . 13 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ( 𝑥𝑧 𝑥𝐴))
30 fvex 6901 . . . . . . . . . . . . . 14 (topGen‘𝐵) ∈ V
31 simpr 486 . . . . . . . . . . . . . 14 ((𝐵𝑉𝐴𝑊) → 𝐴𝑊)
32 uniiun 5060 . . . . . . . . . . . . . . . 16 𝑧 = 𝑥𝑧 𝑥
33 eltg3i 22446 . . . . . . . . . . . . . . . 16 ((𝐵𝑉𝑧𝐵) → 𝑧 ∈ (topGen‘𝐵))
3432, 33eqeltrrid 2839 . . . . . . . . . . . . . . 15 ((𝐵𝑉𝑧𝐵) → 𝑥𝑧 𝑥 ∈ (topGen‘𝐵))
3534adantlr 714 . . . . . . . . . . . . . 14 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → 𝑥𝑧 𝑥 ∈ (topGen‘𝐵))
36 elrestr 17370 . . . . . . . . . . . . . 14 (((topGen‘𝐵) ∈ V ∧ 𝐴𝑊 𝑥𝑧 𝑥 ∈ (topGen‘𝐵)) → ( 𝑥𝑧 𝑥𝐴) ∈ ((topGen‘𝐵) ↾t 𝐴))
3730, 31, 35, 36mp3an2ani 1469 . . . . . . . . . . . . 13 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ( 𝑥𝑧 𝑥𝐴) ∈ ((topGen‘𝐵) ↾t 𝐴))
3829, 37eqeltrd 2834 . . . . . . . . . . . 12 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) ∈ ((topGen‘𝐵) ↾t 𝐴))
39 unieq 4918 . . . . . . . . . . . . 13 (𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) → 𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧))
4039eleq1d 2819 . . . . . . . . . . . 12 (𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) → ( 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴) ↔ ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) ∈ ((topGen‘𝐵) ↾t 𝐴)))
4138, 40syl5ibrcom 246 . . . . . . . . . . 11 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → (𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) → 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4241expimpd 455 . . . . . . . . . 10 ((𝐵𝑉𝐴𝑊) → ((𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)) → 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4342exlimdv 1937 . . . . . . . . 9 ((𝐵𝑉𝐴𝑊) → (∃𝑧(𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)) → 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4443adantr 482 . . . . . . . 8 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → (∃𝑧(𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)) → 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4519, 44mpd 15 . . . . . . 7 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴))
46 eleq1 2822 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴) ↔ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4745, 46syl5ibrcom 246 . . . . . 6 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → (𝑥 = 𝑦𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4847expimpd 455 . . . . 5 ((𝐵𝑉𝐴𝑊) → ((𝑦 ⊆ (𝐵t 𝐴) ∧ 𝑥 = 𝑦) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4948exlimdv 1937 . . . 4 ((𝐵𝑉𝐴𝑊) → (∃𝑦(𝑦 ⊆ (𝐵t 𝐴) ∧ 𝑥 = 𝑦) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
503, 49biimtrid 241 . . 3 ((𝐵𝑉𝐴𝑊) → (𝑥 ∈ (topGen‘(𝐵t 𝐴)) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
5150ssrdv 3987 . 2 ((𝐵𝑉𝐴𝑊) → (topGen‘(𝐵t 𝐴)) ⊆ ((topGen‘𝐵) ↾t 𝐴))
52 restval 17368 . . . 4 (((topGen‘𝐵) ∈ V ∧ 𝐴𝑊) → ((topGen‘𝐵) ↾t 𝐴) = ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)))
5330, 31, 52sylancr 588 . . 3 ((𝐵𝑉𝐴𝑊) → ((topGen‘𝐵) ↾t 𝐴) = ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)))
54 eltg3 22447 . . . . . . . 8 (𝐵𝑉 → (𝑤 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧𝐵𝑤 = 𝑧)))
5554adantr 482 . . . . . . 7 ((𝐵𝑉𝐴𝑊) → (𝑤 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧𝐵𝑤 = 𝑧)))
5632ineq1i 4207 . . . . . . . . . . . 12 ( 𝑧𝐴) = ( 𝑥𝑧 𝑥𝐴)
5756, 28eqtr4i 2764 . . . . . . . . . . 11 ( 𝑧𝐴) = 𝑥𝑧 (𝑥𝐴)
58 simplll 774 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → 𝐵𝑉)
59 simpllr 775 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → 𝐴𝑊)
60 simpr 486 . . . . . . . . . . . . . . . . 17 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → 𝑧𝐵)
6160sselda 3981 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → 𝑥𝐵)
62 elrestr 17370 . . . . . . . . . . . . . . . 16 ((𝐵𝑉𝐴𝑊𝑥𝐵) → (𝑥𝐴) ∈ (𝐵t 𝐴))
6358, 59, 61, 62syl3anc 1372 . . . . . . . . . . . . . . 15 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → (𝑥𝐴) ∈ (𝐵t 𝐴))
6463fmpttd 7110 . . . . . . . . . . . . . 14 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → (𝑥𝑧 ↦ (𝑥𝐴)):𝑧⟶(𝐵t 𝐴))
6564frnd 6722 . . . . . . . . . . . . 13 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ran (𝑥𝑧 ↦ (𝑥𝐴)) ⊆ (𝐵t 𝐴))
66 eltg3i 22446 . . . . . . . . . . . . 13 (((𝐵t 𝐴) ∈ V ∧ ran (𝑥𝑧 ↦ (𝑥𝐴)) ⊆ (𝐵t 𝐴)) → ran (𝑥𝑧 ↦ (𝑥𝐴)) ∈ (topGen‘(𝐵t 𝐴)))
671, 65, 66sylancr 588 . . . . . . . . . . . 12 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ran (𝑥𝑧 ↦ (𝑥𝐴)) ∈ (topGen‘(𝐵t 𝐴)))
6826, 67eqeltrid 2838 . . . . . . . . . . 11 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → 𝑥𝑧 (𝑥𝐴) ∈ (topGen‘(𝐵t 𝐴)))
6957, 68eqeltrid 2838 . . . . . . . . . 10 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ( 𝑧𝐴) ∈ (topGen‘(𝐵t 𝐴)))
70 ineq1 4204 . . . . . . . . . . 11 (𝑤 = 𝑧 → (𝑤𝐴) = ( 𝑧𝐴))
7170eleq1d 2819 . . . . . . . . . 10 (𝑤 = 𝑧 → ((𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴)) ↔ ( 𝑧𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7269, 71syl5ibrcom 246 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → (𝑤 = 𝑧 → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7372expimpd 455 . . . . . . . 8 ((𝐵𝑉𝐴𝑊) → ((𝑧𝐵𝑤 = 𝑧) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7473exlimdv 1937 . . . . . . 7 ((𝐵𝑉𝐴𝑊) → (∃𝑧(𝑧𝐵𝑤 = 𝑧) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7555, 74sylbid 239 . . . . . 6 ((𝐵𝑉𝐴𝑊) → (𝑤 ∈ (topGen‘𝐵) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7675imp 408 . . . . 5 (((𝐵𝑉𝐴𝑊) ∧ 𝑤 ∈ (topGen‘𝐵)) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴)))
7776fmpttd 7110 . . . 4 ((𝐵𝑉𝐴𝑊) → (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)):(topGen‘𝐵)⟶(topGen‘(𝐵t 𝐴)))
7877frnd 6722 . . 3 ((𝐵𝑉𝐴𝑊) → ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)) ⊆ (topGen‘(𝐵t 𝐴)))
7953, 78eqsstrd 4019 . 2 ((𝐵𝑉𝐴𝑊) → ((topGen‘𝐵) ↾t 𝐴) ⊆ (topGen‘(𝐵t 𝐴)))
8051, 79eqssd 3998 1 ((𝐵𝑉𝐴𝑊) → (topGen‘(𝐵t 𝐴)) = ((topGen‘𝐵) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wral 3062  Vcvv 3475  cin 3946  wss 3947   cuni 4907   ciun 4996  cmpt 5230  ran crn 5676  cres 5677  cima 5678  Fun wfun 6534   Fn wfn 6535  cfv 6540  (class class class)co 7404  t crest 17362  topGenctg 17379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-rest 17364  df-topgen 17385
This theorem is referenced by:  resttop  22646  ordtrest2  22690  2ndcrest  22940  txrest  23117  xkoptsub  23140  xrtgioo  24304  ordtrest2NEW  32841  ptrest  36425
  Copyright terms: Public domain W3C validator