Let $f : X \to Y$ be an D467: Injective map such that

(i) | $g$ is a D458: Submap of $f$ |

Then $g$ is an D467: Injective map.

Result R399
on D467: Injective map

Injectivity is hereditary

Formulation 0

Let $f : X \to Y$ be an D467: Injective map such that

(i) | $g$ is a D458: Submap of $f$ |

Then $g$ is an D467: Injective map.

Proofs

Let $f : X \to Y$ be an D467: Injective map such that

(i) | $g$ is a D458: Submap of $f$ |

Suppose that $g$ has a domain set $E$ and a codomain set $F$. If $E$ is the empty set, then $g$ is injective due to R2752: Empty map is injection, so we may assume that $E$ is nonempty. Let $x, y \in E$ such that $g(x) = g(y)$. Since $g$ is a submap of $f$, then $f(x) = g(x) = g(y) = f(y)$. Since $f$ is an injection, it follows that $x = y$. Since $x, y \in E$ were arbitrary, $g$ is an injection. $\square$