Implications

We have the following indirect implication of form equivalence classes:

138-k \(\Rightarrow\) 138-k
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
138-k \(\Rightarrow\) 138-k

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
138-k:

Suppose \(k\in\omega\).  If \(f\) is a partial map from \(k\times Y\)  onto \(k\times X\) (that is, the domain is a subset of \(k\times Y\)), then there are partitions \(X = \bigcup_{i \le k} X_{i}\)  and \(Y = \bigcup_{i \le k} Y_{i}\) of \(X\) and \(Y\) such that \(f\) maps \(\bigcup_{i \le k} (\{i\} \times Y_{i})\) onto \(\bigcup^{}_{i \le k} (\{i\} \times X_{i})\).

138-k:

Suppose \(k\in\omega\).  If \(f\) is a partial map from \(k\times Y\)  onto \(k\times X\) (that is, the domain is a subset of \(k\times Y\)), then there are partitions \(X = \bigcup_{i \le k} X_{i}\)  and \(Y = \bigcup_{i \le k} Y_{i}\) of \(X\) and \(Y\) such that \(f\) maps \(\bigcup_{i \le k} (\{i\} \times Y_{i})\) onto \(\bigcup^{}_{i \le k} (\{i\} \times X_{i})\).

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