Metamath Proof Explorer


Theorem idinxpss

Description: Two ways to say that an intersection of the identity relation with a Cartesian product is a subclass. (Contributed by Peter Mazsa, 16-Jul-2019)

Ref Expression
Assertion idinxpss
|- ( ( _I i^i ( A X. B ) ) C_ R <-> A. x e. A A. y e. B ( x = y -> x R y ) )

Proof

Step Hyp Ref Expression
1 inxpss
 |-  ( ( _I i^i ( A X. B ) ) C_ R <-> A. x e. A A. y e. B ( x _I y -> x R y ) )
2 ideqg
 |-  ( y e. _V -> ( x _I y <-> x = y ) )
3 2 elv
 |-  ( x _I y <-> x = y )
4 3 imbi1i
 |-  ( ( x _I y -> x R y ) <-> ( x = y -> x R y ) )
5 4 2ralbii
 |-  ( A. x e. A A. y e. B ( x _I y -> x R y ) <-> A. x e. A A. y e. B ( x = y -> x R y ) )
6 1 5 bitri
 |-  ( ( _I i^i ( A X. B ) ) C_ R <-> A. x e. A A. y e. B ( x = y -> x R y ) )