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r - Matching information from different dataframes and filtering out redundant columns

In the first dataframe are site names that have been matched up with corresponding grid cells. These grid cells have unique column and row numbers. Here's an example of the first dataframe:

Site <- as.data.frame(c("Site.A","Site.B","Site.C"))
Row <- as.data.frame(c(1,2,3))
Column <- as.data.frame(c(5,4,3))
df1 <- cbind(Site,Row, Column)
colnames(df1) <- c("Site","Row","Column")

In a separate dataframe, I have separate information from all possible grid cells. An example:

eg1 <- rbind(c(1,2,3,4,5),c(5,4,3,2,1))
eg2 <- as.data.frame(matrix(sample(0:50, 15*10, replace=TRUE), ncol=5))
df2 <- rbind(eg1,eg2)
rownames(df2)[1:2] <- c("Row","Column")

What I would like to do is to filter columns in df2 so that they only have grid cells with columns and rows in df1. I would then need to match each site name with its corresponding grid cell. An example output of what I need.

Output <- df2[,1:3]
colnames(Output) <- c("Site.A","Site.B","Site.C")
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One solution is as follows:

df2[, (df2['Row', ] %in% df1$Row) & (df2['Column', ] %in% df1$Column)]

Head of the output is as follows:

       V1 V2 V3
Row     1  2  3
Column  5  4  3
3      49 29 34
4      45 42 18
5       9 15 45
6      34 35 19

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