Metamath Proof Explorer


Theorem relpeq3

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

Ref Expression
Assertion relpeq3 Could not format assertion : No typesetting found for |- ( S = T -> ( H RelPres R , S ( A , B ) <-> H RelPres R , T ( A , B ) ) ) with typecode |-

Proof

Step Hyp Ref Expression
1 breq S = T H x S H y H x T H y
2 1 imbi2d S = T x R y H x S H y x R y H x T H y
3 2 2ralbidv S = T x A y A x R y H x S H y x A y A x R y H x T H y
4 3 anbi2d S = T H : A B x A y A x R y H x S H y H : A B x A y A x R y H x T 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 R , T ( A , B ) <-> ( H : A --> B /\ A. x e. A A. y e. A ( x R y -> ( H ` x ) T ( H ` y ) ) ) ) : No typesetting found for |- ( H RelPres R , T ( A , B ) <-> ( H : A --> B /\ A. x e. A A. y e. A ( x R y -> ( H ` x ) T ( H ` y ) ) ) ) with typecode |-
7 4 5 6 3bitr4g Could not format ( S = T -> ( H RelPres R , S ( A , B ) <-> H RelPres R , T ( A , B ) ) ) : No typesetting found for |- ( S = T -> ( H RelPres R , S ( A , B ) <-> H RelPres R , T ( A , B ) ) ) with typecode |-