  This paper is a technical continuation of ``Natural Axiom Schemata Extending
ZFC. Truth in the Universe?'' In that paper we argue that $CIFS$ is a natural
axiom schema for the universe of sets. In particular it is a natural closure
condition on $V$ and a natural generalization of $IFS(L).$ Here we shall prove
the consistency of $ZFC\ +\ CIFS$ relative to the existence of a transitive
model of $ZFC$ using the compactness theorem together with a class forcing.
