Union of Right-Total Relations is Right-Total

Theorem

Let $S_1, S_2, T_1, T_2$ be sets or classes.

Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be right-total relations.

Then $\RR_1 \cup \RR_2$ is right-total.

Define the predicates $L$ and $R$ by:

$\map L X \iff \text {$X$ is left-total}$

$\map R X \iff \text {$X$ is right-total}$

$\ds \map R {\RR_1} \land \map R {\RR_2}$

$\leadsto$

$\ds \map L {\RR_1^{-1} } \land \map L {\RR_2^{-1} }$

$\ds $

$\leadsto$

$\ds \map L {\RR_1^{-1} \cup \RR_2^{-1} }$

$\ds $

$\leadsto$

$\ds \map L {\paren {\RR_1 \cup \RR_2}^{-1} }$

$\ds $

$\leadsto$

$\ds \map R {\RR_1 \cup \RR_2}$

$\blacksquare$