Metamath Proof Explorer

Theorem satf0sucom

Description: The satisfaction predicate as function over wff codes in the empty model with an empty binary relation at a successor of _om . (Contributed by AV, 14-Sep-2023)

Ref Expression
Assertion satf0sucom 
|- ( N e. suc _om -> ( ( (/) Sat (/) ) ` N ) = ( rec ( ( f e. _V |-> ( f u. { <. x , y >. | ( y = (/) /\ E. u e. f ( E. v e. f x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } ) ) , { <. x , y >. | ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) } ) ` N ) )

Proof

Step Hyp Ref Expression
1 satf0 
 |-  ( (/) Sat (/) ) = ( rec ( ( f e. _V |-> ( f u. { <. x , y >. | ( y = (/) /\ E. u e. f ( E. v e. f x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } ) ) , { <. x , y >. | ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) } ) |` suc _om )
2 1 fveq1i 
 |-  ( ( (/) Sat (/) ) ` N ) = ( ( rec ( ( f e. _V |-> ( f u. { <. x , y >. | ( y = (/) /\ E. u e. f ( E. v e. f x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } ) ) , { <. x , y >. | ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) } ) |` suc _om ) ` N )
3 fvres 
 |-  ( N e. suc _om -> ( ( rec ( ( f e. _V |-> ( f u. { <. x , y >. | ( y = (/) /\ E. u e. f ( E. v e. f x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } ) ) , { <. x , y >. | ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) } ) |` suc _om ) ` N ) = ( rec ( ( f e. _V |-> ( f u. { <. x , y >. | ( y = (/) /\ E. u e. f ( E. v e. f x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } ) ) , { <. x , y >. | ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) } ) ` N ) )
4 2 3 syl5eq 
 |-  ( N e. suc _om -> ( ( (/) Sat (/) ) ` N ) = ( rec ( ( f e. _V |-> ( f u. { <. x , y >. | ( y = (/) /\ E. u e. f ( E. v e. f x = ( ( 1st ` u ) |g ( 1st ` v ) ) \/ E. i e. _om x = A.g i ( 1st ` u ) ) ) } ) ) , { <. x , y >. | ( y = (/) /\ E. i e. _om E. j e. _om x = ( i e.g j ) ) } ) ` N ) )