Metamath Proof Explorer


Theorem fsovfvd

Description: Value of the operator, ( A O B ) , which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, A and B , when applied to function F . (Contributed by RP, 25-Apr-2021)

Ref Expression
Hypotheses fsovd.fs
|- O = ( a e. _V , b e. _V |-> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) )
fsovd.a
|- ( ph -> A e. V )
fsovd.b
|- ( ph -> B e. W )
fsovfvd.g
|- G = ( A O B )
fsovfvd.f
|- ( ph -> F e. ( ~P B ^m A ) )
Assertion fsovfvd
|- ( ph -> ( G ` F ) = ( y e. B |-> { x e. A | y e. ( F ` x ) } ) )

Proof

Step Hyp Ref Expression
1 fsovd.fs
 |-  O = ( a e. _V , b e. _V |-> ( f e. ( ~P b ^m a ) |-> ( y e. b |-> { x e. a | y e. ( f ` x ) } ) ) )
2 fsovd.a
 |-  ( ph -> A e. V )
3 fsovd.b
 |-  ( ph -> B e. W )
4 fsovfvd.g
 |-  G = ( A O B )
5 fsovfvd.f
 |-  ( ph -> F e. ( ~P B ^m A ) )
6 1 2 3 fsovd
 |-  ( ph -> ( A O B ) = ( f e. ( ~P B ^m A ) |-> ( y e. B |-> { x e. A | y e. ( f ` x ) } ) ) )
7 4 6 syl5eq
 |-  ( ph -> G = ( f e. ( ~P B ^m A ) |-> ( y e. B |-> { x e. A | y e. ( f ` x ) } ) ) )
8 fveq1
 |-  ( f = F -> ( f ` x ) = ( F ` x ) )
9 8 eleq2d
 |-  ( f = F -> ( y e. ( f ` x ) <-> y e. ( F ` x ) ) )
10 9 rabbidv
 |-  ( f = F -> { x e. A | y e. ( f ` x ) } = { x e. A | y e. ( F ` x ) } )
11 10 mpteq2dv
 |-  ( f = F -> ( y e. B |-> { x e. A | y e. ( f ` x ) } ) = ( y e. B |-> { x e. A | y e. ( F ` x ) } ) )
12 11 adantl
 |-  ( ( ph /\ f = F ) -> ( y e. B |-> { x e. A | y e. ( f ` x ) } ) = ( y e. B |-> { x e. A | y e. ( F ` x ) } ) )
13 3 mptexd
 |-  ( ph -> ( y e. B |-> { x e. A | y e. ( F ` x ) } ) e. _V )
14 7 12 5 13 fvmptd
 |-  ( ph -> ( G ` F ) = ( y e. B |-> { x e. A | y e. ( F ` x ) } ) )