This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem pjclii

Description: Closure of a projection in its subspace. (Contributed by NM, 30-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypotheses pjcli.1 ⊢ 𝐻 ∈ Cℋ
pjcli.2 ⊢ 𝐴 ∈ ℋ
Assertion pjclii ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐻

Proof

Step Hyp Ref Expression
1 pjcli.1 ⊢ 𝐻 ∈ Cℋ
2 pjcli.2 ⊢ 𝐴 ∈ ℋ
3 1 pjcli ⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐻 )
4 2 3 ax-mp ⊢ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐻