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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 ( 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 ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | cA | ⊢ 𝐴 | |
| 1 | cB | ⊢ 𝐵 | |
| 2 | 0 1 | cres | ⊢ ( 𝐴 ↾ 𝐵 ) |
| 3 | cvv | ⊢ V | |
| 4 | 1 3 | cxp | ⊢ ( 𝐵 × V ) |
| 5 | 0 4 | cin | ⊢ ( 𝐴 ∩ ( 𝐵 × V ) ) |
| 6 | 2 5 | wceq | ⊢ ( 𝐴 ↾ 𝐵 ) = ( 𝐴 ∩ ( 𝐵 × V ) ) |