Metamath Proof Explorer


Theorem relpeq2

Description: Equality theorem for relation-preserving functions. (Contributed by Eric Schmidt, 11-Oct-2025)

Ref Expression
Assertion relpeq2 Could not format assertion : No typesetting found for |- ( R = T -> ( H RelPres R , S ( A , B ) <-> H RelPres T , S ( A , B ) ) ) with typecode |-

Proof

Step Hyp Ref Expression
1 breq R = T x R y x T y
2 1 imbi1d R = T x R y H x S H y x T y H x S H y
3 2 2ralbidv R = T x A y A x R y H x S H y x A y A x T y H x S H y
4 3 anbi2d R = T H : A B x A y A x R y H x S H y H : A B x A y A x T y H x S H y
5 df-relp Could not format ( H RelPres R , S ( A , B ) <-> ( H : A --> B /\ A. x e. A A. y e. A ( x R y -> ( H ` x ) S ( H ` y ) ) ) ) : No typesetting found for |- ( H RelPres R , S ( A , B ) <-> ( H : A --> B /\ A. x e. A A. y e. A ( x R y -> ( H ` x ) S ( H ` y ) ) ) ) with typecode |-
6 df-relp Could not format ( H RelPres T , S ( A , B ) <-> ( H : A --> B /\ A. x e. A A. y e. A ( x T y -> ( H ` x ) S ( H ` y ) ) ) ) : No typesetting found for |- ( H RelPres T , S ( A , B ) <-> ( H : A --> B /\ A. x e. A A. y e. A ( x T y -> ( H ` x ) S ( H ` y ) ) ) ) with typecode |-
7 4 5 6 3bitr4g Could not format ( R = T -> ( H RelPres R , S ( A , B ) <-> H RelPres T , S ( A , B ) ) ) : No typesetting found for |- ( R = T -> ( H RelPres R , S ( A , B ) <-> H RelPres T , S ( A , B ) ) ) with typecode |-