Metamath Proof Explorer


Theorem relpeq5

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

Ref Expression
Assertion relpeq5 Could not format assertion : No typesetting found for |- ( B = C -> ( H RelPres R , S ( A , B ) <-> H RelPres R , S ( A , C ) ) ) with typecode |-

Proof

Step Hyp Ref Expression
1 feq3 B = C H : A B H : A C
2 1 anbi1d B = C H : A B x A y A x R y H x S H y H : A C x A y A x R y H x S H y
3 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 |-
4 df-relp Could not format ( H RelPres R , S ( A , C ) <-> ( H : A --> C /\ 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 , C ) <-> ( H : A --> C /\ A. x e. A A. y e. A ( x R y -> ( H ` x ) S ( H ` y ) ) ) ) with typecode |-
5 2 3 4 3bitr4g Could not format ( B = C -> ( H RelPres R , S ( A , B ) <-> H RelPres R , S ( A , C ) ) ) : No typesetting found for |- ( B = C -> ( H RelPres R , S ( A , B ) <-> H RelPres R , S ( A , C ) ) ) with typecode |-