Theorem extssr

Description: Property of subset relation, see also extid , extep and the comment of df-ssr . (Contributed by Peter Mazsa, 10-Jul-2019)

	Ref	Expression
Assertion	extssr	|- ( ( A e. V /\ B e. W ) -> ( [ A ] `' _S = [ B ] `' _S <-> A = B ) )

Proof

Step	Hyp	Ref	Expression
1		brssr	|- ( A e. V -> ( x _S A <-> x C_ A ) )
2		brssr	|- ( B e. W -> ( x _S B <-> x C_ B ) )
3	1 2	bi2bian9	|- ( ( A e. V /\ B e. W ) -> ( ( x _S A <-> x _S B ) <-> ( x C_ A <-> x C_ B ) ) )
4	3	albidv	|- ( ( A e. V /\ B e. W ) -> ( A. x ( x _S A <-> x _S B ) <-> A. x ( x C_ A <-> x C_ B ) ) )
5		relssr	|- Rel _S
6		releccnveq	|- ( ( Rel _S /\ Rel _S ) -> ( [ A ] `' _S = [ B ] `' _S <-> A. x ( x _S A <-> x _S B ) ) )
7	5 5 6	mp2an	|- ( [ A ] `' _S = [ B ] `' _S <-> A. x ( x _S A <-> x _S B ) )
8		ssext	|- ( A = B <-> A. x ( x C_ A <-> x C_ B ) )
9	4 7 8	3bitr4g	|- ( ( A e. V /\ B e. W ) -> ( [ A ] `' _S = [ B ] `' _S <-> A = B ) )