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Theorem tgrest 21011
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 6718 . . . . 5 (𝐵t 𝐴) ∈ V
2 eltg3 20814 . . . . 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 805 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → 𝐵𝑉)
5 funmpt 5964 . . . . . . . . . 10 Fun (𝑥𝐵 ↦ (𝑥𝐴))
65a1i 11 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → Fun (𝑥𝐵 ↦ (𝑥𝐴)))
7 restval 16134 . . . . . . . . . . . 12 ((𝐵𝑉𝐴𝑊) → (𝐵t 𝐴) = ran (𝑥𝐵 ↦ (𝑥𝐴)))
87sseq2d 3666 . . . . . . . . . . 11 ((𝐵𝑉𝐴𝑊) → (𝑦 ⊆ (𝐵t 𝐴) ↔ 𝑦 ⊆ ran (𝑥𝐵 ↦ (𝑥𝐴))))
98biimpa 500 . . . . . . . . . 10 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → 𝑦 ⊆ ran (𝑥𝐵 ↦ (𝑥𝐴)))
10 vex 3234 . . . . . . . . . . . . 13 𝑥 ∈ V
1110inex1 4832 . . . . . . . . . . . 12 (𝑥𝐴) ∈ V
1211rgenw 2953 . . . . . . . . . . 11 𝑥𝐵 (𝑥𝐴) ∈ V
13 eqid 2651 . . . . . . . . . . . 12 (𝑥𝐵 ↦ (𝑥𝐴)) = (𝑥𝐵 ↦ (𝑥𝐴))
1413fnmpt 6058 . . . . . . . . . . 11 (∀𝑥𝐵 (𝑥𝐴) ∈ V → (𝑥𝐵 ↦ (𝑥𝐴)) Fn 𝐵)
15 fnima 6048 . . . . . . . . . . 11 ((𝑥𝐵 ↦ (𝑥𝐴)) Fn 𝐵 → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵) = ran (𝑥𝐵 ↦ (𝑥𝐴)))
1612, 14, 15mp2b 10 . . . . . . . . . 10 ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵) = ran (𝑥𝐵 ↦ (𝑥𝐴))
179, 16syl6sseqr 3685 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → 𝑦 ⊆ ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵))
18 ssimaexg 6303 . . . . . . . . 9 ((𝐵𝑉 ∧ Fun (𝑥𝐵 ↦ (𝑥𝐴)) ∧ 𝑦 ⊆ ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵)) → ∃𝑧(𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)))
194, 6, 17, 18syl3anc 1366 . . . . . . . 8 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → ∃𝑧(𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)))
20 df-ima 5156 . . . . . . . . . . . . . . . . 17 ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ran ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧)
21 resmpt 5484 . . . . . . . . . . . . . . . . . . 19 (𝑧𝐵 → ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧) = (𝑥𝑧 ↦ (𝑥𝐴)))
2221adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧) = (𝑥𝑧 ↦ (𝑥𝐴)))
2322rneqd 5385 . . . . . . . . . . . . . . . . 17 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ran ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧) = ran (𝑥𝑧 ↦ (𝑥𝐴)))
2420, 23syl5eq 2697 . . . . . . . . . . . . . . . 16 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ran (𝑥𝑧 ↦ (𝑥𝐴)))
2524unieqd 4478 . . . . . . . . . . . . . . 15 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ran (𝑥𝑧 ↦ (𝑥𝐴)))
2611dfiun3 5412 . . . . . . . . . . . . . . 15 𝑥𝑧 (𝑥𝐴) = ran (𝑥𝑧 ↦ (𝑥𝐴))
2725, 26syl6eqr 2703 . . . . . . . . . . . . . 14 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = 𝑥𝑧 (𝑥𝐴))
28 iunin1 4617 . . . . . . . . . . . . . 14 𝑥𝑧 (𝑥𝐴) = ( 𝑥𝑧 𝑥𝐴)
2927, 28syl6eq 2701 . . . . . . . . . . . . 13 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ( 𝑥𝑧 𝑥𝐴))
30 fvex 6239 . . . . . . . . . . . . . . 15 (topGen‘𝐵) ∈ V
3130a1i 11 . . . . . . . . . . . . . 14 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → (topGen‘𝐵) ∈ V)
32 simpr 476 . . . . . . . . . . . . . . 15 ((𝐵𝑉𝐴𝑊) → 𝐴𝑊)
3332adantr 480 . . . . . . . . . . . . . 14 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → 𝐴𝑊)
34 uniiun 4605 . . . . . . . . . . . . . . . 16 𝑧 = 𝑥𝑧 𝑥
35 eltg3i 20813 . . . . . . . . . . . . . . . 16 ((𝐵𝑉𝑧𝐵) → 𝑧 ∈ (topGen‘𝐵))
3634, 35syl5eqelr 2735 . . . . . . . . . . . . . . 15 ((𝐵𝑉𝑧𝐵) → 𝑥𝑧 𝑥 ∈ (topGen‘𝐵))
3736adantlr 751 . . . . . . . . . . . . . 14 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → 𝑥𝑧 𝑥 ∈ (topGen‘𝐵))
38 elrestr 16136 . . . . . . . . . . . . . 14 (((topGen‘𝐵) ∈ V ∧ 𝐴𝑊 𝑥𝑧 𝑥 ∈ (topGen‘𝐵)) → ( 𝑥𝑧 𝑥𝐴) ∈ ((topGen‘𝐵) ↾t 𝐴))
3931, 33, 37, 38syl3anc 1366 . . . . . . . . . . . . 13 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ( 𝑥𝑧 𝑥𝐴) ∈ ((topGen‘𝐵) ↾t 𝐴))
4029, 39eqeltrd 2730 . . . . . . . . . . . 12 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) ∈ ((topGen‘𝐵) ↾t 𝐴))
41 unieq 4476 . . . . . . . . . . . . 13 (𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) → 𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧))
4241eleq1d 2715 . . . . . . . . . . . 12 (𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) → ( 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴) ↔ ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) ∈ ((topGen‘𝐵) ↾t 𝐴)))
4340, 42syl5ibrcom 237 . . . . . . . . . . 11 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → (𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) → 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4443expimpd 628 . . . . . . . . . 10 ((𝐵𝑉𝐴𝑊) → ((𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)) → 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4544exlimdv 1901 . . . . . . . . 9 ((𝐵𝑉𝐴𝑊) → (∃𝑧(𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)) → 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4645adantr 480 . . . . . . . 8 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → (∃𝑧(𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)) → 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4719, 46mpd 15 . . . . . . 7 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴))
48 eleq1 2718 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴) ↔ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4947, 48syl5ibrcom 237 . . . . . 6 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → (𝑥 = 𝑦𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
5049expimpd 628 . . . . 5 ((𝐵𝑉𝐴𝑊) → ((𝑦 ⊆ (𝐵t 𝐴) ∧ 𝑥 = 𝑦) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
5150exlimdv 1901 . . . 4 ((𝐵𝑉𝐴𝑊) → (∃𝑦(𝑦 ⊆ (𝐵t 𝐴) ∧ 𝑥 = 𝑦) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
523, 51syl5bi 232 . . 3 ((𝐵𝑉𝐴𝑊) → (𝑥 ∈ (topGen‘(𝐵t 𝐴)) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
5352ssrdv 3642 . 2 ((𝐵𝑉𝐴𝑊) → (topGen‘(𝐵t 𝐴)) ⊆ ((topGen‘𝐵) ↾t 𝐴))
54 restval 16134 . . . 4 (((topGen‘𝐵) ∈ V ∧ 𝐴𝑊) → ((topGen‘𝐵) ↾t 𝐴) = ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)))
5530, 32, 54sylancr 696 . . 3 ((𝐵𝑉𝐴𝑊) → ((topGen‘𝐵) ↾t 𝐴) = ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)))
56 eltg3 20814 . . . . . . . 8 (𝐵𝑉 → (𝑤 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧𝐵𝑤 = 𝑧)))
5756adantr 480 . . . . . . 7 ((𝐵𝑉𝐴𝑊) → (𝑤 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧𝐵𝑤 = 𝑧)))
5834ineq1i 3843 . . . . . . . . . . . 12 ( 𝑧𝐴) = ( 𝑥𝑧 𝑥𝐴)
5958, 28eqtr4i 2676 . . . . . . . . . . 11 ( 𝑧𝐴) = 𝑥𝑧 (𝑥𝐴)
60 simplll 813 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → 𝐵𝑉)
61 simpllr 815 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → 𝐴𝑊)
62 simpr 476 . . . . . . . . . . . . . . . . 17 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → 𝑧𝐵)
6362sselda 3636 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → 𝑥𝐵)
64 elrestr 16136 . . . . . . . . . . . . . . . 16 ((𝐵𝑉𝐴𝑊𝑥𝐵) → (𝑥𝐴) ∈ (𝐵t 𝐴))
6560, 61, 63, 64syl3anc 1366 . . . . . . . . . . . . . . 15 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → (𝑥𝐴) ∈ (𝐵t 𝐴))
66 eqid 2651 . . . . . . . . . . . . . . 15 (𝑥𝑧 ↦ (𝑥𝐴)) = (𝑥𝑧 ↦ (𝑥𝐴))
6765, 66fmptd 6425 . . . . . . . . . . . . . 14 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → (𝑥𝑧 ↦ (𝑥𝐴)):𝑧⟶(𝐵t 𝐴))
68 frn 6091 . . . . . . . . . . . . . 14 ((𝑥𝑧 ↦ (𝑥𝐴)):𝑧⟶(𝐵t 𝐴) → ran (𝑥𝑧 ↦ (𝑥𝐴)) ⊆ (𝐵t 𝐴))
6967, 68syl 17 . . . . . . . . . . . . 13 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ran (𝑥𝑧 ↦ (𝑥𝐴)) ⊆ (𝐵t 𝐴))
70 eltg3i 20813 . . . . . . . . . . . . 13 (((𝐵t 𝐴) ∈ V ∧ ran (𝑥𝑧 ↦ (𝑥𝐴)) ⊆ (𝐵t 𝐴)) → ran (𝑥𝑧 ↦ (𝑥𝐴)) ∈ (topGen‘(𝐵t 𝐴)))
711, 69, 70sylancr 696 . . . . . . . . . . . 12 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ran (𝑥𝑧 ↦ (𝑥𝐴)) ∈ (topGen‘(𝐵t 𝐴)))
7226, 71syl5eqel 2734 . . . . . . . . . . 11 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → 𝑥𝑧 (𝑥𝐴) ∈ (topGen‘(𝐵t 𝐴)))
7359, 72syl5eqel 2734 . . . . . . . . . 10 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ( 𝑧𝐴) ∈ (topGen‘(𝐵t 𝐴)))
74 ineq1 3840 . . . . . . . . . . 11 (𝑤 = 𝑧 → (𝑤𝐴) = ( 𝑧𝐴))
7574eleq1d 2715 . . . . . . . . . 10 (𝑤 = 𝑧 → ((𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴)) ↔ ( 𝑧𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7673, 75syl5ibrcom 237 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → (𝑤 = 𝑧 → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7776expimpd 628 . . . . . . . 8 ((𝐵𝑉𝐴𝑊) → ((𝑧𝐵𝑤 = 𝑧) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7877exlimdv 1901 . . . . . . 7 ((𝐵𝑉𝐴𝑊) → (∃𝑧(𝑧𝐵𝑤 = 𝑧) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7957, 78sylbid 230 . . . . . 6 ((𝐵𝑉𝐴𝑊) → (𝑤 ∈ (topGen‘𝐵) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
8079imp 444 . . . . 5 (((𝐵𝑉𝐴𝑊) ∧ 𝑤 ∈ (topGen‘𝐵)) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴)))
81 eqid 2651 . . . . 5 (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)) = (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴))
8280, 81fmptd 6425 . . . 4 ((𝐵𝑉𝐴𝑊) → (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)):(topGen‘𝐵)⟶(topGen‘(𝐵t 𝐴)))
83 frn 6091 . . . 4 ((𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)):(topGen‘𝐵)⟶(topGen‘(𝐵t 𝐴)) → ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)) ⊆ (topGen‘(𝐵t 𝐴)))
8482, 83syl 17 . . 3 ((𝐵𝑉𝐴𝑊) → ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)) ⊆ (topGen‘(𝐵t 𝐴)))
8555, 84eqsstrd 3672 . 2 ((𝐵𝑉𝐴𝑊) → ((topGen‘𝐵) ↾t 𝐴) ⊆ (topGen‘(𝐵t 𝐴)))
8653, 85eqssd 3653 1 ((𝐵𝑉𝐴𝑊) → (topGen‘(𝐵t 𝐴)) = ((topGen‘𝐵) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wral 2941  Vcvv 3231  cin 3606  wss 3607   cuni 4468   ciun 4552  cmpt 4762  ran crn 5144  cres 5145  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  t crest 16128  topGenctg 16145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-rest 16130  df-topgen 16151
This theorem is referenced by:  resttop  21012  ordtrest2  21056  2ndcrest  21305  txrest  21482  xkoptsub  21505  xrtgioo  22656  ordtrest2NEW  30097  ptrest  33538
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