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Metamath Proof Explorer

Theorem rescnvcnv

Description: The restriction of the double converse of a class. (Contributed by NM, 8-Apr-2007) (Proof shortened by Andrew Salmon, 27-Aug-2011)

		Ref	Expression
	Assertion	rescnvcnv	$⊢ {{A^{-1}}^{-1} ↾}_{B} = {A ↾}_{B}$

Step	Hyp	Ref	Expression
1		cnvcnv2	$⊢ {A^{-1}}^{-1} = {A ↾}_{V}$
2	1	reseq1i	$⊢ {{A^{-1}}^{-1} ↾}_{B} = {({A ↾}_{V}) ↾}_{B}$
3		resres	$⊢ {({A ↾}_{V}) ↾}_{B} = {A ↾}_{(V \cap B)}$
4		ssv	$⊢ B \subseteq V$
5		sseqin2	$⊢ B \subseteq V \leftrightarrow V \cap B = B$
6	4 5	mpbi	$⊢ V \cap B = B$
7	6	reseq2i	$⊢ {A ↾}_{(V \cap B)} = {A ↾}_{B}$
8	2 3 7	3eqtri	$⊢ {{A^{-1}}^{-1} ↾}_{B} = {A ↾}_{B}$