Definition·D017
Identity function
The function sending every element of a set to itself.
In words
The identity function on A is the set of pairs p in the product of A with itself for which there is some x with x in A and p equal to the ordered pair of x with x, so that the identity sends each x to x itself for every x in A.
Rests onno axioms yet
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
Exists by Separation inside the product
, and it is a function: each
is paired with exactly one value, namely
itself, by L01. It is the do-nothing bijection witnessing that every set is equinumerous with itself, and the neutral element for composition.