Metamath Proof Explorer

Theorem satefv

Description: The simplified satisfaction predicate as function over wff codes in the model M at the code U . (Contributed by AV, 30-Oct-2023)

Ref Expression
Assertion satefv 
|- ( ( M e. V /\ U e. W ) -> ( M SatE U ) = ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) )

Proof

Step Hyp Ref Expression
1 df-sate 
 |-  SatE = ( m e. _V , u e. _V |-> ( ( ( m Sat ( _E i^i ( m X. m ) ) ) ` _om ) ` u ) )
2 1 a1i 
 |-  ( ( M e. V /\ U e. W ) -> SatE = ( m e. _V , u e. _V |-> ( ( ( m Sat ( _E i^i ( m X. m ) ) ) ` _om ) ` u ) ) )
3 id 
 |-  ( m = M -> m = M )
4 3 sqxpeqd 
 |-  ( m = M -> ( m X. m ) = ( M X. M ) )
5 4 ineq2d 
 |-  ( m = M -> ( _E i^i ( m X. m ) ) = ( _E i^i ( M X. M ) ) )
6 3 5 oveq12d 
 |-  ( m = M -> ( m Sat ( _E i^i ( m X. m ) ) ) = ( M Sat ( _E i^i ( M X. M ) ) ) )
7 6 fveq1d 
 |-  ( m = M -> ( ( m Sat ( _E i^i ( m X. m ) ) ) ` _om ) = ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) )
8 7 adantr 
 |-  ( ( m = M /\ u = U ) -> ( ( m Sat ( _E i^i ( m X. m ) ) ) ` _om ) = ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) )
9 simpr 
 |-  ( ( m = M /\ u = U ) -> u = U )
10 8 9 fveq12d 
 |-  ( ( m = M /\ u = U ) -> ( ( ( m Sat ( _E i^i ( m X. m ) ) ) ` _om ) ` u ) = ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) )
11 10 adantl 
 |-  ( ( ( M e. V /\ U e. W ) /\ ( m = M /\ u = U ) ) -> ( ( ( m Sat ( _E i^i ( m X. m ) ) ) ` _om ) ` u ) = ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) )
12 elex 
 |-  ( M e. V -> M e. _V )
13 12 adantr 
 |-  ( ( M e. V /\ U e. W ) -> M e. _V )
14 elex 
 |-  ( U e. W -> U e. _V )
15 14 adantl 
 |-  ( ( M e. V /\ U e. W ) -> U e. _V )
16 fvexd 
 |-  ( ( M e. V /\ U e. W ) -> ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) e. _V )
17 2 11 13 15 16 ovmpod 
 |-  ( ( M e. V /\ U e. W ) -> ( M SatE U ) = ( ( ( M Sat ( _E i^i ( M X. M ) ) ) ` _om ) ` U ) )