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Metamath Proof Explorer

Theorem isorel

Description: An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004)

Ref Expression
Assertion isorel 
|- ( ( H Isom 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-isom 
 |-  ( H Isom R , S ( A , B ) <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x R y <-> ( H ` x ) S ( H ` y ) ) ) )
2 1 simprbi 
 |-  ( H Isom 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 bibi12d 
 |-  ( 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 bibi12d 
 |-  ( 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 Isom R , S ( A , B ) /\ ( C e. A /\ D e. A ) ) -> ( C R D <-> ( H ` C ) S ( H ` D ) ) )