Metamath Proof Explorer

Theorem bj-elid4

Description: Characterization of the elements of _I . (Contributed by BJ, 22-Jun-2019)

Ref Expression
Assertion bj-elid4 
|- ( A e. ( V X. W ) -> ( A e. _I <-> ( 1st ` A ) = ( 2nd ` A ) ) )

Proof

Step Hyp Ref Expression
1 1st2nd2 
 |-  ( A e. ( V X. W ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2 eleq1 
 |-  ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. -> ( A e. _I <-> <. ( 1st ` A ) , ( 2nd ` A ) >. e. _I ) )
3 2 adantl 
 |-  ( ( A e. ( V X. W ) /\ A = <. ( 1st ` A ) , ( 2nd ` A ) >. ) -> ( A e. _I <-> <. ( 1st ` A ) , ( 2nd ` A ) >. e. _I ) )
4 fvex 
 |-  ( 2nd ` A ) e. _V
5 4 inex2 
 |-  ( ( 1st ` A ) i^i ( 2nd ` A ) ) e. _V
6 bj-opelid 
 |-  ( ( ( 1st ` A ) i^i ( 2nd ` A ) ) e. _V -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. _I <-> ( 1st ` A ) = ( 2nd ` A ) ) )
7 5 6 mp1i 
 |-  ( ( A e. ( V X. W ) /\ A = <. ( 1st ` A ) , ( 2nd ` A ) >. ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. _I <-> ( 1st ` A ) = ( 2nd ` A ) ) )
8 3 7 bitrd 
 |-  ( ( A e. ( V X. W ) /\ A = <. ( 1st ` A ) , ( 2nd ` A ) >. ) -> ( A e. _I <-> ( 1st ` A ) = ( 2nd ` A ) ) )
9 1 8 mpdan 
 |-  ( A e. ( V X. W ) -> ( A e. _I <-> ( 1st ` A ) = ( 2nd ` A ) ) )