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Theorem tgrest 20773
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 6577 . . . . 5 (𝐵t 𝐴) ∈ V
2 eltg3 20577 . . . . 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 786 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → 𝐵𝑉)
5 funmpt 5840 . . . . . . . . . 10 Fun (𝑥𝐵 ↦ (𝑥𝐴))
65a1i 11 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → Fun (𝑥𝐵 ↦ (𝑥𝐴)))
7 restval 15910 . . . . . . . . . . . 12 ((𝐵𝑉𝐴𝑊) → (𝐵t 𝐴) = ran (𝑥𝐵 ↦ (𝑥𝐴)))
87sseq2d 3596 . . . . . . . . . . 11 ((𝐵𝑉𝐴𝑊) → (𝑦 ⊆ (𝐵t 𝐴) ↔ 𝑦 ⊆ ran (𝑥𝐵 ↦ (𝑥𝐴))))
98biimpa 500 . . . . . . . . . 10 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → 𝑦 ⊆ ran (𝑥𝐵 ↦ (𝑥𝐴)))
10 vex 3176 . . . . . . . . . . . . 13 𝑥 ∈ V
1110inex1 4727 . . . . . . . . . . . 12 (𝑥𝐴) ∈ V
1211rgenw 2908 . . . . . . . . . . 11 𝑥𝐵 (𝑥𝐴) ∈ V
13 eqid 2610 . . . . . . . . . . . 12 (𝑥𝐵 ↦ (𝑥𝐴)) = (𝑥𝐵 ↦ (𝑥𝐴))
1413fnmpt 5933 . . . . . . . . . . 11 (∀𝑥𝐵 (𝑥𝐴) ∈ V → (𝑥𝐵 ↦ (𝑥𝐴)) Fn 𝐵)
15 fnima 5923 . . . . . . . . . . 11 ((𝑥𝐵 ↦ (𝑥𝐴)) Fn 𝐵 → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵) = ran (𝑥𝐵 ↦ (𝑥𝐴)))
1612, 14, 15mp2b 10 . . . . . . . . . 10 ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵) = ran (𝑥𝐵 ↦ (𝑥𝐴))
179, 16syl6sseqr 3615 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → 𝑦 ⊆ ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵))
18 ssimaexg 6174 . . . . . . . . 9 ((𝐵𝑉 ∧ Fun (𝑥𝐵 ↦ (𝑥𝐴)) ∧ 𝑦 ⊆ ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝐵)) → ∃𝑧(𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)))
194, 6, 17, 18syl3anc 1318 . . . . . . . 8 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → ∃𝑧(𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)))
20 df-ima 5051 . . . . . . . . . . . . . . . . 17 ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ran ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧)
21 resmpt 5369 . . . . . . . . . . . . . . . . . . 19 (𝑧𝐵 → ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧) = (𝑥𝑧 ↦ (𝑥𝐴)))
2221adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧) = (𝑥𝑧 ↦ (𝑥𝐴)))
2322rneqd 5274 . . . . . . . . . . . . . . . . 17 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ran ((𝑥𝐵 ↦ (𝑥𝐴)) ↾ 𝑧) = ran (𝑥𝑧 ↦ (𝑥𝐴)))
2420, 23syl5eq 2656 . . . . . . . . . . . . . . . 16 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ran (𝑥𝑧 ↦ (𝑥𝐴)))
2524unieqd 4382 . . . . . . . . . . . . . . 15 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ran (𝑥𝑧 ↦ (𝑥𝐴)))
2611dfiun3 5301 . . . . . . . . . . . . . . 15 𝑥𝑧 (𝑥𝐴) = ran (𝑥𝑧 ↦ (𝑥𝐴))
2725, 26syl6eqr 2662 . . . . . . . . . . . . . 14 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = 𝑥𝑧 (𝑥𝐴))
28 iunin1 4521 . . . . . . . . . . . . . 14 𝑥𝑧 (𝑥𝐴) = ( 𝑥𝑧 𝑥𝐴)
2927, 28syl6eq 2660 . . . . . . . . . . . . 13 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) = ( 𝑥𝑧 𝑥𝐴))
30 fvex 6113 . . . . . . . . . . . . . . 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 4509 . . . . . . . . . . . . . . . 16 𝑧 = 𝑥𝑧 𝑥
35 eltg3i 20576 . . . . . . . . . . . . . . . 16 ((𝐵𝑉𝑧𝐵) → 𝑧 ∈ (topGen‘𝐵))
3634, 35syl5eqelr 2693 . . . . . . . . . . . . . . 15 ((𝐵𝑉𝑧𝐵) → 𝑥𝑧 𝑥 ∈ (topGen‘𝐵))
3736adantlr 747 . . . . . . . . . . . . . 14 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → 𝑥𝑧 𝑥 ∈ (topGen‘𝐵))
38 elrestr 15912 . . . . . . . . . . . . . 14 (((topGen‘𝐵) ∈ V ∧ 𝐴𝑊 𝑥𝑧 𝑥 ∈ (topGen‘𝐵)) → ( 𝑥𝑧 𝑥𝐴) ∈ ((topGen‘𝐵) ↾t 𝐴))
3931, 33, 37, 38syl3anc 1318 . . . . . . . . . . . . 13 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ( 𝑥𝑧 𝑥𝐴) ∈ ((topGen‘𝐵) ↾t 𝐴))
4029, 39eqeltrd 2688 . . . . . . . . . . . 12 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) ∈ ((topGen‘𝐵) ↾t 𝐴))
41 unieq 4380 . . . . . . . . . . . . 13 (𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) → 𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧))
4241eleq1d 2672 . . . . . . . . . . . 12 (𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) → ( 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴) ↔ ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) ∈ ((topGen‘𝐵) ↾t 𝐴)))
4340, 42syl5ibrcom 236 . . . . . . . . . . 11 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → (𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧) → 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4443expimpd 627 . . . . . . . . . 10 ((𝐵𝑉𝐴𝑊) → ((𝑧𝐵𝑦 = ((𝑥𝐵 ↦ (𝑥𝐴)) “ 𝑧)) → 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4544exlimdv 1848 . . . . . . . . 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 2676 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴) ↔ 𝑦 ∈ ((topGen‘𝐵) ↾t 𝐴)))
4947, 48syl5ibrcom 236 . . . . . 6 (((𝐵𝑉𝐴𝑊) ∧ 𝑦 ⊆ (𝐵t 𝐴)) → (𝑥 = 𝑦𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
5049expimpd 627 . . . . 5 ((𝐵𝑉𝐴𝑊) → ((𝑦 ⊆ (𝐵t 𝐴) ∧ 𝑥 = 𝑦) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
5150exlimdv 1848 . . . 4 ((𝐵𝑉𝐴𝑊) → (∃𝑦(𝑦 ⊆ (𝐵t 𝐴) ∧ 𝑥 = 𝑦) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
523, 51syl5bi 231 . . 3 ((𝐵𝑉𝐴𝑊) → (𝑥 ∈ (topGen‘(𝐵t 𝐴)) → 𝑥 ∈ ((topGen‘𝐵) ↾t 𝐴)))
5352ssrdv 3574 . 2 ((𝐵𝑉𝐴𝑊) → (topGen‘(𝐵t 𝐴)) ⊆ ((topGen‘𝐵) ↾t 𝐴))
54 restval 15910 . . . 4 (((topGen‘𝐵) ∈ V ∧ 𝐴𝑊) → ((topGen‘𝐵) ↾t 𝐴) = ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)))
5530, 32, 54sylancr 694 . . 3 ((𝐵𝑉𝐴𝑊) → ((topGen‘𝐵) ↾t 𝐴) = ran (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)))
56 eltg3 20577 . . . . . . . 8 (𝐵𝑉 → (𝑤 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧𝐵𝑤 = 𝑧)))
5756adantr 480 . . . . . . 7 ((𝐵𝑉𝐴𝑊) → (𝑤 ∈ (topGen‘𝐵) ↔ ∃𝑧(𝑧𝐵𝑤 = 𝑧)))
5834ineq1i 3772 . . . . . . . . . . . 12 ( 𝑧𝐴) = ( 𝑥𝑧 𝑥𝐴)
5958, 28eqtr4i 2635 . . . . . . . . . . 11 ( 𝑧𝐴) = 𝑥𝑧 (𝑥𝐴)
60 simplll 794 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → 𝐵𝑉)
61 simpllr 795 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → 𝐴𝑊)
62 simpr 476 . . . . . . . . . . . . . . . . 17 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → 𝑧𝐵)
6362sselda 3568 . . . . . . . . . . . . . . . 16 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → 𝑥𝐵)
64 elrestr 15912 . . . . . . . . . . . . . . . 16 ((𝐵𝑉𝐴𝑊𝑥𝐵) → (𝑥𝐴) ∈ (𝐵t 𝐴))
6560, 61, 63, 64syl3anc 1318 . . . . . . . . . . . . . . 15 ((((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) ∧ 𝑥𝑧) → (𝑥𝐴) ∈ (𝐵t 𝐴))
66 eqid 2610 . . . . . . . . . . . . . . 15 (𝑥𝑧 ↦ (𝑥𝐴)) = (𝑥𝑧 ↦ (𝑥𝐴))
6765, 66fmptd 6292 . . . . . . . . . . . . . 14 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → (𝑥𝑧 ↦ (𝑥𝐴)):𝑧⟶(𝐵t 𝐴))
68 frn 5966 . . . . . . . . . . . . . 14 ((𝑥𝑧 ↦ (𝑥𝐴)):𝑧⟶(𝐵t 𝐴) → ran (𝑥𝑧 ↦ (𝑥𝐴)) ⊆ (𝐵t 𝐴))
6967, 68syl 17 . . . . . . . . . . . . 13 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ran (𝑥𝑧 ↦ (𝑥𝐴)) ⊆ (𝐵t 𝐴))
70 eltg3i 20576 . . . . . . . . . . . . 13 (((𝐵t 𝐴) ∈ V ∧ ran (𝑥𝑧 ↦ (𝑥𝐴)) ⊆ (𝐵t 𝐴)) → ran (𝑥𝑧 ↦ (𝑥𝐴)) ∈ (topGen‘(𝐵t 𝐴)))
711, 69, 70sylancr 694 . . . . . . . . . . . 12 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ran (𝑥𝑧 ↦ (𝑥𝐴)) ∈ (topGen‘(𝐵t 𝐴)))
7226, 71syl5eqel 2692 . . . . . . . . . . 11 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → 𝑥𝑧 (𝑥𝐴) ∈ (topGen‘(𝐵t 𝐴)))
7359, 72syl5eqel 2692 . . . . . . . . . 10 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → ( 𝑧𝐴) ∈ (topGen‘(𝐵t 𝐴)))
74 ineq1 3769 . . . . . . . . . . 11 (𝑤 = 𝑧 → (𝑤𝐴) = ( 𝑧𝐴))
7574eleq1d 2672 . . . . . . . . . 10 (𝑤 = 𝑧 → ((𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴)) ↔ ( 𝑧𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7673, 75syl5ibrcom 236 . . . . . . . . 9 (((𝐵𝑉𝐴𝑊) ∧ 𝑧𝐵) → (𝑤 = 𝑧 → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7776expimpd 627 . . . . . . . 8 ((𝐵𝑉𝐴𝑊) → ((𝑧𝐵𝑤 = 𝑧) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7877exlimdv 1848 . . . . . . 7 ((𝐵𝑉𝐴𝑊) → (∃𝑧(𝑧𝐵𝑤 = 𝑧) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
7957, 78sylbid 229 . . . . . 6 ((𝐵𝑉𝐴𝑊) → (𝑤 ∈ (topGen‘𝐵) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴))))
8079imp 444 . . . . 5 (((𝐵𝑉𝐴𝑊) ∧ 𝑤 ∈ (topGen‘𝐵)) → (𝑤𝐴) ∈ (topGen‘(𝐵t 𝐴)))
81 eqid 2610 . . . . 5 (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)) = (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴))
8280, 81fmptd 6292 . . . 4 ((𝐵𝑉𝐴𝑊) → (𝑤 ∈ (topGen‘𝐵) ↦ (𝑤𝐴)):(topGen‘𝐵)⟶(topGen‘(𝐵t 𝐴)))
83 frn 5966 . . . 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 3602 . 2 ((𝐵𝑉𝐴𝑊) → ((topGen‘𝐵) ↾t 𝐴) ⊆ (topGen‘(𝐵t 𝐴)))
8653, 85eqssd 3585 1 ((𝐵𝑉𝐴𝑊) → (topGen‘(𝐵t 𝐴)) = ((topGen‘𝐵) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wex 1695  wcel 1977  wral 2896  Vcvv 3173  cin 3539  wss 3540   cuni 4372   ciun 4455  cmpt 4643  ran crn 5039  cres 5040  cima 5041  Fun wfun 5798   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  t crest 15904  topGenctg 15921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-rest 15906  df-topgen 15927
This theorem is referenced by:  resttop  20774  ordtrest2  20818  2ndcrest  21067  txrest  21244  xkoptsub  21267  xrtgioo  22417  ordtrest2NEW  29297  ptrest  32578
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