The algorithm for proven and young animals (APY) efficiently computes an approximated inverse of the genomic relationship matrix, by dividing genotyped animals in the so-called core and noncore animals. The APY leads to computationally feasible single-step genomic Best Linear Unbiased Prediction (ssGBLUP) with a large number of genotyped animals and was successfully applied to real single-breed or line datasets. This study aimed to assess the quality of genomic estimated breeding values (GEBV) when using the APY (GEBVAPY), in comparison to GEBV when using the directly inverted genomic relationship matrix (GEBVDIRECT), for situations based on crossbreeding schemes, including F1 and F2 crosses, such as the ones for pigs and chickens. Based on simulations of a 3-way crossbreeding program, we compared different approximated inverses of a genomic relationship matrix, by varying the size and the composition of the core group. We showed that GEBVAPY were accurate approximations of GEBVDIRECT for multivariate ssGBLUP involving different breeds and their crosses. GEBVAPY as accurate as GEBVDIRECT were obtained when the core groups included animals from different breed compositions and when the core groups had a size between the numbers of the largest eigenvalues explaining 98% and 99% of the variation in the raw genomic relationship matrix.