Metamath Proof Explorer


Definition df-res

Description: Define the restriction of a class. Definition 6.6(1) of TakeutiZaring p. 24. For example, the expression ` ( exp |`RR ) (used in reeff1 ) means "the exponential function e to the x, but the exponent x must be in the reals" ( df-ef defines the exponential function, which normally allows the exponent to be a complex number). Another example is that ( F = { <. 2 , 6 >. , <. 3 , 9 >. } ` /\ B = { 1 , 2 } ) -> ( F |`B ) = { <. 2 , 6 >. } ( ex-res ). We do not introduce a special syntax for the corestriction of a class: it will be expressed either as the intersection ( A i^i ( _V X. B ) ) or as the converse of the restricted converse (see cnvrescnv ). (Contributed by NM, 2-Aug-1994)

Ref Expression
Assertion df-res A B = A B × V

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA class A
1 cB class B
2 0 1 cres class A B
3 cvv class V
4 1 3 cxp class B × V
5 0 4 cin class A B × V
6 2 5 wceq wff A B = A B × V