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Metamath Proof Explorer

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 ) )