Full Idea
Axiom of Replacement Scheme: ∀x ∈ A ∃!y φ(x,y) → ∃Y ∀X ∈ A ∃y ∈ Y φ(x,y). That is, any function from a set A will produce another set Y.
Gist of Idea
Replacement: ∀x∈A ∃!y φ(x,y) → ∃Y ∀X∈A ∃y∈Y φ(x,y)
Source
Kenneth Kunen (Set Theory [1980], §1.6)
Book Reference
Kunen,Kenneth: 'Set Theory: Introduction to Independence Proofs' [North-Holland 1980], p.12
Related Ideas
Idea 15933 Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine]
Idea 15945 Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine]