Histone demethylase that demethylates 'Lys-4' of histone H3, thereby playing a central role in histone code. Does not demethylate histone H3 'Lys-9' or H3 'Lys-27'. Demethylates trimethylated, dimethylated and monomethylated H3 'Lys-4'. Acts as a transcriptional corepressor (By similarity). 3 2-oxoglutarate + N(6),N(6),N(6)-trimethyl-L-lysyl(4)-[histone H3] + 3 O2 = 3 CO2 + 3 formaldehyde + L-lysyl(4)-[histone H3] + 3 succinate Binds 1 Fe(2+) ion per subunit. Both the JmjC domain and the JmjN domain are required for enzymatic activity. The 2 first PHD-type zinc finger domains are required for transcription repression activity. Belongs to the JARID1 histone demethylase family.