Theorem reltpos

Description: The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015)

	Ref	Expression
Assertion	reltpos	$$⊢\mathrm{Rel}tposF$$

Proof

Step	Hyp	Ref	Expression
1		tposssxp	$$⊢tposF\subseteq \left({\mathrm{dom}F}^{-1}\cup \left\{\varnothing \right\}\right)\times \mathrm{ran}F$$
2		relxp	$$⊢\mathrm{Rel}\left(\left({\mathrm{dom}F}^{-1}\cup \left\{\varnothing \right\}\right)\times \mathrm{ran}F\right)$$
3		relss	$$⊢tposF\subseteq \left({\mathrm{dom}F}^{-1}\cup \left\{\varnothing \right\}\right)\times \mathrm{ran}F\to \left(\mathrm{Rel}\left(\left({\mathrm{dom}F}^{-1}\cup \left\{\varnothing \right\}\right)\times \mathrm{ran}F\right)\to \mathrm{Rel}tposF\right)$$
4	1 2 3	mp2	$$⊢\mathrm{Rel}tposF$$