Definition·D018
Composition of functions
Apply one function, then another: the result is again a function.
for functions
and
, so that
.
In words
The composition of f followed by g is the set of pairs p in the product of A and C for which there are x and z with p the ordered pair of x and z and z the result of applying g after f to x, so that the composite sends x to g of f(x): first f, then g.
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
The defining set exists by Separation inside
(product). That
really is a function
, and that composition preserves injectivity, surjectivity and bijectivity, is L02. Composition is associative,
, since both sides send
to
; this associativity is the prototype for the associativity demanded of a group operation.
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