Metamath Proof Explorer


Theorem metcld2

Description: A subset of a metric space is closed iff every convergent sequence on it converges to a point in the subset. Theorem 1.4-6(b) of Kreyszig p. 30. (Contributed by Mario Carneiro, 1-May-2014)

Ref Expression
Hypothesis metcld.2 J = MetOpen D
Assertion metcld2 D ∞Met X S X S Clsd J t J S S

Proof

Step Hyp Ref Expression
1 metcld.2 J = MetOpen D
2 1 metcld D ∞Met X S X S Clsd J x f f : S f t J x x S
3 19.23v f f : S f t J x x S f f : S f t J x x S
4 vex x V
5 4 elima2 x t J S f f S f t J x
6 id S X S X
7 elfvdm D ∞Met X X dom ∞Met
8 ssexg S X X dom ∞Met S V
9 6 7 8 syl2anr D ∞Met X S X S V
10 nnex V
11 elmapg S V V f S f : S
12 9 10 11 sylancl D ∞Met X S X f S f : S
13 12 anbi1d D ∞Met X S X f S f t J x f : S f t J x
14 13 exbidv D ∞Met X S X f f S f t J x f f : S f t J x
15 5 14 bitr2id D ∞Met X S X f f : S f t J x x t J S
16 15 imbi1d D ∞Met X S X f f : S f t J x x S x t J S x S
17 3 16 syl5bb D ∞Met X S X f f : S f t J x x S x t J S x S
18 17 albidv D ∞Met X S X x f f : S f t J x x S x x t J S x S
19 dfss2 t J S S x x t J S x S
20 18 19 bitr4di D ∞Met X S X x f f : S f t J x x S t J S S
21 2 20 bitrd D ∞Met X S X S Clsd J t J S S