Step |
Hyp |
Ref |
Expression |
1 |
|
psdval.s |
|- S = ( I mPwSer R ) |
2 |
|
psdval.b |
|- B = ( Base ` S ) |
3 |
|
psdval.d |
|- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin } |
4 |
|
psdval.x |
|- ( ph -> X e. I ) |
5 |
|
psdval.f |
|- ( ph -> F e. B ) |
6 |
|
fveq1 |
|- ( f = F -> ( f ` ( k oF + ( y e. I |-> if ( y = X , 1 , 0 ) ) ) ) = ( F ` ( k oF + ( y e. I |-> if ( y = X , 1 , 0 ) ) ) ) ) |
7 |
6
|
oveq2d |
|- ( f = F -> ( ( ( k ` X ) + 1 ) ( .g ` R ) ( f ` ( k oF + ( y e. I |-> if ( y = X , 1 , 0 ) ) ) ) ) = ( ( ( k ` X ) + 1 ) ( .g ` R ) ( F ` ( k oF + ( y e. I |-> if ( y = X , 1 , 0 ) ) ) ) ) ) |
8 |
7
|
mpteq2dv |
|- ( f = F -> ( k e. D |-> ( ( ( k ` X ) + 1 ) ( .g ` R ) ( f ` ( k oF + ( y e. I |-> if ( y = X , 1 , 0 ) ) ) ) ) ) = ( k e. D |-> ( ( ( k ` X ) + 1 ) ( .g ` R ) ( F ` ( k oF + ( y e. I |-> if ( y = X , 1 , 0 ) ) ) ) ) ) ) |
9 |
|
reldmpsr |
|- Rel dom mPwSer |
10 |
9 1 2
|
elbasov |
|- ( F e. B -> ( I e. _V /\ R e. _V ) ) |
11 |
5 10
|
syl |
|- ( ph -> ( I e. _V /\ R e. _V ) ) |
12 |
11
|
simpld |
|- ( ph -> I e. _V ) |
13 |
11
|
simprd |
|- ( ph -> R e. _V ) |
14 |
1 2 3 12 13 4
|
psdfval |
|- ( ph -> ( ( I mPSDer R ) ` X ) = ( f e. B |-> ( k e. D |-> ( ( ( k ` X ) + 1 ) ( .g ` R ) ( f ` ( k oF + ( y e. I |-> if ( y = X , 1 , 0 ) ) ) ) ) ) ) ) |
15 |
|
ovex |
|- ( NN0 ^m I ) e. _V |
16 |
3 15
|
rabex2 |
|- D e. _V |
17 |
16
|
mptex |
|- ( k e. D |-> ( ( ( k ` X ) + 1 ) ( .g ` R ) ( F ` ( k oF + ( y e. I |-> if ( y = X , 1 , 0 ) ) ) ) ) ) e. _V |
18 |
17
|
a1i |
|- ( ph -> ( k e. D |-> ( ( ( k ` X ) + 1 ) ( .g ` R ) ( F ` ( k oF + ( y e. I |-> if ( y = X , 1 , 0 ) ) ) ) ) ) e. _V ) |
19 |
8 14 5 18
|
fvmptd4 |
|- ( ph -> ( ( ( I mPSDer R ) ` X ) ` F ) = ( k e. D |-> ( ( ( k ` X ) + 1 ) ( .g ` R ) ( F ` ( k oF + ( y e. I |-> if ( y = X , 1 , 0 ) ) ) ) ) ) ) |