This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Description: Define the value of an operation. Definition of operation value in Enderton p. 79. Note that the syntax is simply three class expressions in a row bracketed by parentheses. There are no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation F and its arguments A and B - will be useful for proving meaningful theorems. For example, if class F is the operation + and arguments A and B are 3 and 2 , the expression ( 3 + 2 ) can be proved to equal 5 (see 3p2e5 ). This definition is well-defined, although not very meaningful, when classes A and/or B are proper classes (i.e. are not sets); see ovprc1 and ovprc2 . On the other hand, we often find uses for this definition when F is a proper class, such as +o in oav . F is normally equal to a class of nested ordered pairs of the form defined by df-oprab . (Contributed by NM, 28-Feb-1995)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | df-ov | ⊢ ( 𝐴 𝐹 𝐵 ) = ( 𝐹 ‘ 〈 𝐴 , 𝐵 〉 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | cA | ⊢ 𝐴 | |
| 1 | cF | ⊢ 𝐹 | |
| 2 | cB | ⊢ 𝐵 | |
| 3 | 0 2 1 | co | ⊢ ( 𝐴 𝐹 𝐵 ) |
| 4 | 0 2 | cop | ⊢ 〈 𝐴 , 𝐵 〉 |
| 5 | 4 1 | cfv | ⊢ ( 𝐹 ‘ 〈 𝐴 , 𝐵 〉 ) |
| 6 | 3 5 | wceq | ⊢ ( 𝐴 𝐹 𝐵 ) = ( 𝐹 ‘ 〈 𝐴 , 𝐵 〉 ) |