**Description:** A subset of a topology's underlying set is included in its closure.
(Contributed by NM, 22-Feb-2007)

Ref | Expression | ||
---|---|---|---|

Hypothesis | clscld.1 | ⊢ 𝑋 = ∪ 𝐽 | |

Assertion | sscls | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → 𝑆 ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |

Step | Hyp | Ref | Expression |
---|---|---|---|

1 | clscld.1 | ⊢ 𝑋 = ∪ 𝐽 | |

2 | ssintub | ⊢ 𝑆 ⊆ ∩ { 𝑥 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑆 ⊆ 𝑥 } | |

3 | 1 | clsval | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) = ∩ { 𝑥 ∈ ( Clsd ‘ 𝐽 ) ∣ 𝑆 ⊆ 𝑥 } ) |

4 | 2 3 | sseqtrrid | ⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ) → 𝑆 ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |