Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > df-2nd | Structured version Visualization version GIF version |
Description: Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 7849 proves that it does this. For example, (2nd ‘〈3, 4〉) = 4. Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 6136 and op2ndb 6135). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.) |
Ref | Expression |
---|---|
df-2nd | ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c2nd 7839 | . 2 class 2nd | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cvv 3433 | . . 3 class V | |
4 | 2 | cv 1538 | . . . . . 6 class 𝑥 |
5 | 4 | csn 4562 | . . . . 5 class {𝑥} |
6 | 5 | crn 5591 | . . . 4 class ran {𝑥} |
7 | 6 | cuni 4840 | . . 3 class ∪ ran {𝑥} |
8 | 2, 3, 7 | cmpt 5158 | . 2 class (𝑥 ∈ V ↦ ∪ ran {𝑥}) |
9 | 1, 8 | wceq 1539 | 1 wff 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) |
Colors of variables: wff setvar class |
This definition is referenced by: 2ndval 7843 fo2nd 7861 f2ndres 7865 hashf1rn 14076 |
Copyright terms: Public domain | W3C validator |