Metamath Proof Explorer


Theorem rfovfvfvd

Description: Value of the operator, ( A O B ) , which maps between relations and functions for relations between base sets, A and B , relation R , and left element X . (Contributed by RP, 25-Apr-2021)

Ref Expression
Hypotheses rfovd.rf
|- O = ( a e. _V , b e. _V |-> ( r e. ~P ( a X. b ) |-> ( x e. a |-> { y e. b | x r y } ) ) )
rfovd.a
|- ( ph -> A e. V )
rfovd.b
|- ( ph -> B e. W )
rfovfvd.r
|- ( ph -> R e. ~P ( A X. B ) )
rfovfvd.f
|- F = ( A O B )
rfovfvfvd.x
|- ( ph -> X e. A )
rfovfvfvd.g
|- G = ( F ` R )
Assertion rfovfvfvd
|- ( ph -> ( G ` X ) = { y e. B | X R y } )

Proof

Step Hyp Ref Expression
1 rfovd.rf
 |-  O = ( a e. _V , b e. _V |-> ( r e. ~P ( a X. b ) |-> ( x e. a |-> { y e. b | x r y } ) ) )
2 rfovd.a
 |-  ( ph -> A e. V )
3 rfovd.b
 |-  ( ph -> B e. W )
4 rfovfvd.r
 |-  ( ph -> R e. ~P ( A X. B ) )
5 rfovfvd.f
 |-  F = ( A O B )
6 rfovfvfvd.x
 |-  ( ph -> X e. A )
7 rfovfvfvd.g
 |-  G = ( F ` R )
8 1 2 3 4 5 rfovfvd
 |-  ( ph -> ( F ` R ) = ( x e. A |-> { y e. B | x R y } ) )
9 7 8 syl5eq
 |-  ( ph -> G = ( x e. A |-> { y e. B | x R y } ) )
10 breq1
 |-  ( x = X -> ( x R y <-> X R y ) )
11 10 rabbidv
 |-  ( x = X -> { y e. B | x R y } = { y e. B | X R y } )
12 11 adantl
 |-  ( ( ph /\ x = X ) -> { y e. B | x R y } = { y e. B | X R y } )
13 rabexg
 |-  ( B e. W -> { y e. B | X R y } e. _V )
14 3 13 syl
 |-  ( ph -> { y e. B | X R y } e. _V )
15 9 12 6 14 fvmptd
 |-  ( ph -> ( G ` X ) = { y e. B | X R y } )