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Mirrors > Home > ILE Home > Th. List > df-2nd | GIF version |
Description: Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 5774 proves that it does this. For example, (2^{nd} ‘⟨ 3 , 4 ⟩) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 4805 and op2ndb 4804). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.) |
Ref | Expression |
---|---|
df-2nd | ⊢ 2^{nd} = (𝑥 ∈ V ↦ ∪ ran {𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c2nd 5766 | . 2 class 2^{nd} | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cvv 2557 | . . 3 class V | |
4 | 2 | cv 1242 | . . . . . 6 class 𝑥 |
5 | 4 | csn 3375 | . . . . 5 class {𝑥} |
6 | 5 | crn 4346 | . . . 4 class ran {𝑥} |
7 | 6 | cuni 3580 | . . 3 class ∪ ran {𝑥} |
8 | 2, 3, 7 | cmpt 3818 | . 2 class (𝑥 ∈ V ↦ ∪ ran {𝑥}) |
9 | 1, 8 | wceq 1243 | 1 wff 2^{nd} = (𝑥 ∈ V ↦ ∪ ran {𝑥}) |
Colors of variables: wff set class |
This definition is referenced by: 2ndvalg 5770 fo2nd 5785 f2ndres 5787 |
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