Axioms for the theory of proveit.logic.sets.functions.injections

import proveit
# Prepare this notebook for defining the axioms of a theory:
%axioms_notebook # Keep this at the top following 'import proveit'.

from proveit import a, b, f, A, B, fa, fb
from proveit.logic import (And, Forall, Equals, NotEquals, 
                           InSet, Functions, Injections)

%begin axioms

Defining axioms for theory 'proveit.logic.sets.functions.injections'
Subsequent end-of-cell assignments will define axioms
%end_axioms will finalize the definitions

injective_def = Forall(
    (A, B), Forall(
        f, 
        Equals(InSet(f, Injections(A, B)),
               And(InSet(f, Functions(A, B)),
                   Forall((a, b), NotEquals(fa, fb),
                          domain=A, condition=NotEquals(a, b))))
        .with_wrap_after_operator()))

injective_def:

%end axioms

These axioms may now be imported from the theory package: proveit.logic.sets.functions.injections