Theorem cnvtrucl0

Description: The converse of the trivial closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020)

	Ref	Expression
Assertion	cnvtrucl0	$$⊢X\in V\to {\bigcap \left\{x|\left(X\subseteq x\wedge \top \right)\right\}}^{-1}=\bigcap \left\{y|\left(X^{-1}\subseteq y\wedge \top \right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		idd	$$⊢\left(X\in V\wedge x=y^{-1}\cup \left(X\setminus {X^{-1}}^{-1}\right)\right)\to \left(\top \to \top \right)$$
2		idd	$$⊢\left(X\in V\wedge y=x^{-1}\right)\to \left(\top \to \top \right)$$
3		biidd	$$⊢x=X\to \left(\top \leftrightarrow \top \right)$$
4		ssidd	$$⊢X\in V\to X\subseteq X$$
5		elex	$$⊢X\in V\to X\in \mathrm{V}$$
6		trud	$$⊢X\in V\to \top$$
7	1 2 3 4 5 6	clcnvlem	$$⊢X\in V\to {\bigcap \left\{x|\left(X\subseteq x\wedge \top \right)\right\}}^{-1}=\bigcap \left\{y|\left(X^{-1}\subseteq y\wedge \top \right)\right\}$$