Metamath Proof Explorer


Definition df-fi

Description: Function whose value is the class of finite intersections of the elements of the argument. Note that the empty intersection being the universal class, hence a proper class, it cannot be an element of that class. Therefore, the function value is the class of nonempty finite intersections of elements of the argument (see elfi2 ). (Contributed by FL, 27-Apr-2008)

Ref Expression
Assertion df-fi
|- fi = ( x e. _V |-> { z | E. y e. ( ~P x i^i Fin ) z = |^| y } )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cfi
 |-  fi
1 vx
 |-  x
2 cvv
 |-  _V
3 vz
 |-  z
4 vy
 |-  y
5 1 cv
 |-  x
6 5 cpw
 |-  ~P x
7 cfn
 |-  Fin
8 6 7 cin
 |-  ( ~P x i^i Fin )
9 3 cv
 |-  z
10 4 cv
 |-  y
11 10 cint
 |-  |^| y
12 9 11 wceq
 |-  z = |^| y
13 12 4 8 wrex
 |-  E. y e. ( ~P x i^i Fin ) z = |^| y
14 13 3 cab
 |-  { z | E. y e. ( ~P x i^i Fin ) z = |^| y }
15 1 2 14 cmpt
 |-  ( x e. _V |-> { z | E. y e. ( ~P x i^i Fin ) z = |^| y } )
16 0 15 wceq
 |-  fi = ( x e. _V |-> { z | E. y e. ( ~P x i^i Fin ) z = |^| y } )