Does bathing with the intention of purification and ablution suffice for ablution, even if the order of ablution is not adhered to, and the bathing is like regular cleansing bathing?

If the ghusl (full ritual ablution) is performed for the obligatory janabah (major ritual impurity) bath, then it suffices for wudu' (minor ritual ablution). However, if it is for cleansing, cooling down, or a recommended ghusl (such as the Friday ghusl), it does not suffice for wudu' unless one intends to remove the minor impurity and washes the wudu' organs in order during the ghusl; otherwise, wudu' is obligatory after it. The difference is that the obligatory ghusl for major impurity suffices for wudu', whereas the permissible or recommended ghusl is not for major impurity, and thus minor impurity is not included within it, making wudu' necessary to remove the minor impurity. Ibn Baz mentioned that the Friday ghusl or a ghusl for cooling down does not suffice for wudu' even if one intends it, due to the lack of sequence and the absence of a major purification within which the minor one can be subsumed.