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Theorem tgrest 23182
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 7463 . . . . 5 (𝐵t 𝐴) ∈ V
2 eltg3 22984 . . . . 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 767 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → 𝐵𝑉)
5 funmpt 6605 . . . . . . . . . 10 Fun (𝑥𝐵 ↦ (𝑥𝐴))
65a1i 11 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → Fun (𝑥𝐵 ↦ (𝑥𝐴)))
7 restval 17472 . . . . . . . . . . . 12 ((𝐵𝑉𝐴𝑊) → (𝐵t 𝐴) = ran (𝑥𝐵 ↦ (𝑥𝐴)))
87sseq2d 4027 . . . . . . . . . . 11 ((𝐵𝑉𝐴𝑊) → (𝑦 ⊆ (𝐵t 𝐴) ↔ 𝑦 ⊆ ran (𝑥𝐵 ↦ (𝑥𝐴))))
98biimpa 476 . . . . . . . . . 10 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → 𝑦 ⊆ ran (𝑥𝐵 ↦ (𝑥𝐴)))
10 vex 3481 . . . . . . . . . . . . 13 𝑥 ∈ V
1110inex1 5322 . . . . . . . . . . . 12 (𝑥𝐴) ∈ V
1211rgenw 3062 . . . . . . . . . . 11 𝑥𝐵 (𝑥𝐴) ∈ V
13 eqid 2734 . . . . . . . . . . . 12 (𝑥𝐵 ↦ (𝑥𝐴)) = (𝑥𝐵 ↦ (𝑥𝐴))
1413fnmpt 6708 . . . . . . . . . . 11 (∀𝑥𝐵 (𝑥𝐴) ∈ V → (𝑥𝐵 ↦ (𝑥𝐴)) Fn 𝐵)
15 fnima 6698 . . . . . . . . . . 11 ((𝑥𝐵 ↦ (𝑥𝐴)) Fn 𝐵 → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵) = ran (𝑥𝐵 ↦ (𝑥𝐴)))
1612, 14, 15mp2b 10 . . . . . . . . . 10 ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵) = ran (𝑥𝐵 ↦ (𝑥𝐴))
179, 16sseqtrrdi 4046 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → 𝑦 ⊆ ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵))
18 ssimaexg 6994 . . . . . . . . 9 ((𝐵𝑉 ∧ Fun (𝑥𝐵 ↦ (𝑥𝐴)) ∧ 𝑦 ⊆ ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵)) → ∃𝑧(𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)))
194, 6, 17, 18syl3anc 1370 . . . . . . . 8 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → ∃𝑧(𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)))
20 df-ima 5701 . . . . . . . . . . . . . . . . 17 ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ran ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧)
21 resmpt 6056 . . . . . . . . . . . . . . . . . . 19 (𝑧𝐵 → ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧) = (𝑥𝑧 ↦ (𝑥𝐴)))
2221adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧) = (𝑥𝑧 ↦ (𝑥𝐴)))
2322rneqd 5951 . . . . . . . . . . . . . . . . 17 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ran ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧) = ran (𝑥𝑧 ↦ (𝑥𝐴)))
2420, 23eqtrid 2786 . . . . . . . . . . . . . . . 16 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ran (𝑥𝑧 ↦ (𝑥𝐴)))
2524unieqd 4924 . . . . . . . . . . . . . . 15 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ran (𝑥𝑧 ↦ (𝑥𝐴)))
2611dfiun3 5982 . . . . . . . . . . . . . . 15 𝑥𝑧 (𝑥𝐴) = ran (𝑥𝑧 ↦ (𝑥𝐴))
2725, 26eqtr4di 2792 . . . . . . . . . . . . . 14 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = 𝑥𝑧 (𝑥𝐴))
28 iunin1 5076 . . . . . . . . . . . . . 14 𝑥𝑧 (𝑥𝐴) = ( 𝑥𝑧 𝑥𝐴)
2927, 28eqtrdi 2790 . . . . . . . . . . . . 13 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ( 𝑥𝑧 𝑥𝐴))
30 fvex 6919 . . . . . . . . . . . . . 14 (topGen‘𝐵) ∈ V
31 simpr 484 . . . . . . . . . . . . . 14 ((𝐵𝑉𝐴𝑊) → 𝐴𝑊)
32 uniiun 5062 . . . . . . . . . . . . . . . 16 𝑧 = 𝑥𝑧 𝑥
33 eltg3i 22983 . . . . . . . . . . . . . . . 16 ((𝐵𝑉𝑧𝐵) → 𝑧 ∈ (topGen‘𝐵))
3432, 33eqeltrrid 2843 . . . . . . . . . . . . . . 15 ((𝐵𝑉𝑧𝐵) → 𝑥𝑧 𝑥 ∈ (topGen‘𝐵))
3534adantlr 715 . . . . . . . . . . . . . 14 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → 𝑥𝑧 𝑥 ∈ (topGen‘𝐵))
36 elrestr 17474 . . . . . . . . . . . . . 14 (((topGen‘𝐵) ∈ V ∧ 𝐴𝑊 𝑥𝑧 𝑥 ∈ (topGen‘𝐵)) → ( 𝑥𝑧 𝑥𝐴) ∈ ((topGen‘𝐵) ↾t 𝐴))
3730, 31, 35, 36mp3an2ani 1467 . . . . . . . . . . . . 13 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ( 𝑥𝑧 𝑥𝐴) ∈ ((topGen‘𝐵) ↾t 𝐴))
3829, 37eqeltrd 2838 . . . . . . . . . . . 12 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) ∈ ((topGen‘𝐵) ↾t 𝐴))
39 unieq 4922 . . . . . . . . . . . . 13 (𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) → 𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧))
4039eleq1d 2823 . . . . . . . . . . . 12 (𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) → ( 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴) ↔ ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) ∈ ((topGen‘𝐵) ↾t 𝐴)))
4138, 40syl5ibrcom 247 . . . . . . . . . . 11 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → (𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) → 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4241expimpd 453 . . . . . . . . . 10 ((𝐵𝑉𝐴𝑊) → ((𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)) → 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4342exlimdv 1930 . . . . . . . . 9 ((𝐵𝑉𝐴𝑊) → (∃𝑧(𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)) → 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4443adantr 480 . . . . . . . 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 2826 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴) ↔ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4745, 46syl5ibrcom 247 . . . . . 6 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → (𝑥 = 𝑦𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4847expimpd 453 . . . . 5 ((𝐵𝑉𝐴𝑊) → ((𝑦 ⊆ (𝐵t 𝐴) ∧ 𝑥 = 𝑦) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4948exlimdv 1930 . . . 4 ((𝐵𝑉𝐴𝑊) → (∃𝑦(𝑦 ⊆ (𝐵t 𝐴) ∧ 𝑥 = 𝑦) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
503, 49biimtrid 242 . . 3 ((𝐵𝑉𝐴𝑊) → (𝑥 ∈ (topGen‘(𝐵t 𝐴)) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
5150ssrdv 4000 . 2 ((𝐵𝑉𝐴𝑊) → (topGen‘(𝐵t 𝐴)) ⊆ ((topGen‘𝐵) ↾t 𝐴))
52 restval 17472 . . . 4 (((topGen‘𝐵) ∈ V ∧ 𝐴𝑊) → ((topGen‘𝐵) ↾t 𝐴) = ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)))
5330, 31, 52sylancr 587 . . 3 ((𝐵𝑉𝐴𝑊) → ((topGen‘𝐵) ↾t 𝐴) = ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)))
54 eltg3 22984 . . . . . . . 8 (𝐵𝑉 → (𝑤 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧𝐵𝑤 = 𝑧)))
5554adantr 480 . . . . . . 7 ((𝐵𝑉𝐴𝑊) → (𝑤 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧𝐵𝑤 = 𝑧)))
5632ineq1i 4223 . . . . . . . . . . . 12 ( 𝑧𝐴) = ( 𝑥𝑧 𝑥𝐴)
5756, 28eqtr4i 2765 . . . . . . . . . . 11 ( 𝑧𝐴) = 𝑥𝑧 (𝑥𝐴)
58 simplll 775 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → 𝐵𝑉)
59 simpllr 776 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → 𝐴𝑊)
60 simpr 484 . . . . . . . . . . . . . . . . 17 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → 𝑧𝐵)
6160sselda 3994 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → 𝑥𝐵)
62 elrestr 17474 . . . . . . . . . . . . . . . 16 ((𝐵𝑉𝐴𝑊𝑥𝐵) → (𝑥𝐴) ∈ (𝐵t 𝐴))
6358, 59, 61, 62syl3anc 1370 . . . . . . . . . . . . . . 15 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → (𝑥𝐴) ∈ (𝐵t 𝐴))
6463fmpttd 7134 . . . . . . . . . . . . . 14 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → (𝑥𝑧 ↦ (𝑥𝐴)):𝑧⟶(𝐵t 𝐴))
6564frnd 6744 . . . . . . . . . . . . 13 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ran (𝑥𝑧 ↦ (𝑥𝐴)) ⊆ (𝐵t 𝐴))
66 eltg3i 22983 . . . . . . . . . . . . 13 (((𝐵t 𝐴) ∈ V ∧ ran (𝑥𝑧 ↦ (𝑥𝐴)) ⊆ (𝐵t 𝐴)) → ran (𝑥𝑧 ↦ (𝑥𝐴)) ∈ (topGen‘(𝐵t 𝐴)))
671, 65, 66sylancr 587 . . . . . . . . . . . 12 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ran (𝑥𝑧 ↦ (𝑥𝐴)) ∈ (topGen‘(𝐵t 𝐴)))
6826, 67eqeltrid 2842 . . . . . . . . . . 11 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → 𝑥𝑧 (𝑥𝐴) ∈ (topGen‘(𝐵t 𝐴)))
6957, 68eqeltrid 2842 . . . . . . . . . 10 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ( 𝑧𝐴) ∈ (topGen‘(𝐵t 𝐴)))
70 ineq1 4220 . . . . . . . . . . 11 (𝑤 = 𝑧 → (𝑤𝐴) = ( 𝑧𝐴))
7170eleq1d 2823 . . . . . . . . . 10 (𝑤 = 𝑧 → ((𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴)) ↔ ( 𝑧𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7269, 71syl5ibrcom 247 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → (𝑤 = 𝑧 → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7372expimpd 453 . . . . . . . 8 ((𝐵𝑉𝐴𝑊) → ((𝑧𝐵𝑤 = 𝑧) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7473exlimdv 1930 . . . . . . 7 ((𝐵𝑉𝐴𝑊) → (∃𝑧(𝑧𝐵𝑤 = 𝑧) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7555, 74sylbid 240 . . . . . 6 ((𝐵𝑉𝐴𝑊) → (𝑤 ∈ (topGen‘𝐵) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7675imp 406 . . . . 5 (((𝐵𝑉𝐴𝑊) ∧ 𝑤 ∈ (topGen‘𝐵)) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴)))
7776fmpttd 7134 . . . 4 ((𝐵𝑉𝐴𝑊) → (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)):(topGen‘𝐵)⟶(topGen‘(𝐵t 𝐴)))
7877frnd 6744 . . 3 ((𝐵𝑉𝐴𝑊) → ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)) ⊆ (topGen‘(𝐵t 𝐴)))
7953, 78eqsstrd 4033 . 2 ((𝐵𝑉𝐴𝑊) → ((topGen‘𝐵) ↾t 𝐴) ⊆ (topGen‘(𝐵t 𝐴)))
8051, 79eqssd 4012 1 ((𝐵𝑉𝐴𝑊) → (topGen‘(𝐵t 𝐴)) = ((topGen‘𝐵) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wex 1775  wcel 2105  wral 3058  Vcvv 3477  cin 3961  wss 3962   cuni 4911   ciun 4995  cmpt 5230  ran crn 5689  cres 5690  cima 5691  Fun wfun 6556   Fn wfn 6557  cfv 6562  (class class class)co 7430  t crest 17466  topGenctg 17483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-rest 17468  df-topgen 17489
This theorem is referenced by:  resttop  23183  ordtrest2  23227  2ndcrest  23477  txrest  23654  xkoptsub  23677  xrtgioo  24841  ordtrest2NEW  33883  ptrest  37605
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