Definition df-2nd 7996

Description: Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 8004 proves that it does this. For example, (2^nd ‘⟨3, 4⟩) = 4. Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 6234 and op2ndb 6233). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)

Assertion

Ref	Expression
df-2nd	⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})

Detailed syntax breakdown of Definition df-2nd

Step	Hyp	Ref	Expression
1		c2nd 7994	. 2 class 2nd
2		vx	. . 3 setvar 𝑥
3		cvv 3458	. . 3 class V
4	2	cv 1569	. . . . . 6 class 𝑥
5	4	csn 4594	. . . . 5 class {𝑥}
6	5	crn 5667	. . . 4 class ran {𝑥}
7	6	cuni 4877	. . 3 class ∪ ran {𝑥}
8	2, 3, 7	cmpt 5197	. 2 class (𝑥 ∈ V ↦ ∪ ran {𝑥})
9	1, 8	wceq 1570	1 wff 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥})

This definition is used by:  2ndval  7998  fo2nd  8016  f2ndres  8020  hashf1rn  14408