Certain very large cardinals are not created in small forcing extensions

The large cardinal axioms of the title assert, respectively, the existence of a nontrivial elementary embedding j : V λ → V λ , the existence of such a j which is moreover Σ n 1 , and the existence of such a j which extends to an elementary j : V λ + 1 → V λ + 1 . It is known that these axioms are preserved in passing from a ground model to a small forcing extension. In this paper the reverse directions of these preservations are proved. Also the following is shown (and used in the above proofs in place of using a standard fact): if V is a model of ZFC and V [ G ] is a P -generic forcing extension of V , then in V [ G ] , V is definable using the parameter V δ + 1 , where δ = P = + .