Metamath Proof Explorer


Definition df-sate

Description: A simplified version of the satisfaction predicate, using the standard membership relation and eliminating the extra variable n . (Contributed by Mario Carneiro, 14-Jul-2013)

Ref Expression
Assertion df-sate
|- SatE = ( m e. _V , u e. _V |-> ( ( ( m Sat ( _E i^i ( m X. m ) ) ) ` _om ) ` u ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 csate
 |-  SatE
1 vm
 |-  m
2 cvv
 |-  _V
3 vu
 |-  u
4 1 cv
 |-  m
5 csat
 |-  Sat
6 cep
 |-  _E
7 4 4 cxp
 |-  ( m X. m )
8 6 7 cin
 |-  ( _E i^i ( m X. m ) )
9 4 8 5 co
 |-  ( m Sat ( _E i^i ( m X. m ) ) )
10 com
 |-  _om
11 10 9 cfv
 |-  ( ( m Sat ( _E i^i ( m X. m ) ) ) ` _om )
12 3 cv
 |-  u
13 12 11 cfv
 |-  ( ( ( m Sat ( _E i^i ( m X. m ) ) ) ` _om ) ` u )
14 1 3 2 2 13 cmpo
 |-  ( m e. _V , u e. _V |-> ( ( ( m Sat ( _E i^i ( m X. m ) ) ) ` _om ) ` u ) )
15 0 14 wceq
 |-  SatE = ( m e. _V , u e. _V |-> ( ( ( m Sat ( _E i^i ( m X. m ) ) ) ` _om ) ` u ) )