Metamath Proof Explorer

Theorem relprel

Description: A relation-preserving function preserves the relation. (Contributed by Eric Schmidt, 11-Oct-2025)

Ref Expression
Assertion relprel 
|- ( ( H RelPres R , S ( A , B ) /\ ( C e. A /\ D e. A ) ) -> ( C R D -> ( H ` C ) S ( H ` D ) ) )

Proof

Step Hyp Ref Expression
1 df-relp 
 |-  ( 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 ) ) ) )
2 1 simprbi 
 |-  ( H RelPres R , S ( A , B ) -> A. x e. A A. y e. A ( x R y -> ( H ` x ) S ( H ` y ) ) )
3 breq1 
 |-  ( x = C -> ( x R y <-> C R y ) )
4 fveq2 
 |-  ( x = C -> ( H ` x ) = ( H ` C ) )
5 4 breq1d 
 |-  ( x = C -> ( ( H ` x ) S ( H ` y ) <-> ( H ` C ) S ( H ` y ) ) )
6 3 5 imbi12d 
 |-  ( x = C -> ( ( x R y -> ( H ` x ) S ( H ` y ) ) <-> ( C R y -> ( H ` C ) S ( H ` y ) ) ) )
7 breq2 
 |-  ( y = D -> ( C R y <-> C R D ) )
8 fveq2 
 |-  ( y = D -> ( H ` y ) = ( H ` D ) )
9 8 breq2d 
 |-  ( y = D -> ( ( H ` C ) S ( H ` y ) <-> ( H ` C ) S ( H ` D ) ) )
10 7 9 imbi12d 
 |-  ( y = D -> ( ( C R y -> ( H ` C ) S ( H ` y ) ) <-> ( C R D -> ( H ` C ) S ( H ` D ) ) ) )
11 6 10 rspc2v 
 |-  ( ( C e. A /\ D e. A ) -> ( A. x e. A A. y e. A ( x R y -> ( H ` x ) S ( H ` y ) ) -> ( C R D -> ( H ` C ) S ( H ` D ) ) ) )
12 2 11 mpan9 
 |-  ( ( H RelPres R , S ( A , B ) /\ ( C e. A /\ D e. A ) ) -> ( C R D -> ( H ` C ) S ( H ` D ) ) )