Find elements of a vector in a list and collect information in data frame

From a vector with gene ids, I want to identify which sublist contains one or more of the ids and the associated pvalue.
The result should be a list of data frames with a col for each gene ID, a second col indicating if the ID appears as boolean and a third col with the pval of that gene ID)

Thanks to Pauls question i realized that a gene ID might be in several sublists and the pvalue would indicate its specificity. Accordingly, the expected result should be a list for a df for each sublist and a boolean vector indicating if the geneID is present. Sorry for the edit.

example data

   the_list <- list(
  a = data.frame(ids = c("ABB", 'SDG', 'SHD', 'DUR'), pval = c(0.01, 0.03, 0.05, 0.05)),
  b = data.frame(ids =c('DYR' ,'LRH' ,'FPR', 'FUR', 'DCTWE', 'IRN', 'DRB'), pval = c(0.01, 0.03, 0.05, 0.05, 4, 5, 6)),
  c = data.frame(ids =c('SYR' ,'SDT', 'DFN' ,'FRQ' ,'DFRR', 'SDR'), pval = c(0.01, 0.03, 0.05, 0.05, 5, 2))
)

the_vector <- c("ABB", 'FUR', 'DFN')




 expected_result <- list(
  'a' = data.frame(
    gene_ids = c("ABB", 'FUR', 'DFN'),
    in_list = c(T, F, F),
    pval = c(0.01, NA, NA)),
  'b' = data.frame(
    gene_ids = c("ABB", 'FUR', 'DFN'),
    in_list = c(F, T, F),
    pval = c(NA, 0.05, NA)),
  'c' = data.frame(
    gene_ids = c("ABB", 'FUR', 'DFN'),
    in_list = c(F, F, T),
    pval = c(NA,  NA, 0.05))
  )

Base R solution:

# For each list element: 
# res_list => list of data.frames
res_list <- lapply(
  the_list,
  function(x){
    # Resolve the index of the matched elements: 
    # idx => integer vector
    idx <- match(the_vector, x$ids)
    # Resolve the required data.frame: 
    # data.frame => env
    data.frame(
      gene_ids = the_vector, 
      in_list = !(is.na(idx)),
      p_val = x$pval[idx],
      stringsAsFactors = FALSE,
      row.names = NULL
    )
  }
)

Loop through the list of data.frames and merge:

lapply(the_list, function(i){
  res <- merge(data.frame(ids = the_vector), i, all.x = TRUE)
  res$in_list <- !is.na(res$pval)
  res
  })
# $a
#   ids pval in_list
# 1 ABB 0.01    TRUE
# 2 DFN   NA   FALSE
# 3 FUR   NA   FALSE
# 
# $b
#   ids pval in_list
# 1 ABB   NA   FALSE
# 2 DFN   NA   FALSE
# 3 FUR 0.05    TRUE
# 
# $c
#   ids pval in_list
# 1 ABB   NA   FALSE
# 2 DFN 0.05    TRUE
# 3 FUR   NA   FALSE

dplyr::bind_rows(the_list, .id = "in_list") %>%
  filter(ids %in% the_vector)

gives

  in_list ids pval
1       a ABB 0.01
2       b FUR 0.05
3       c DFN 0.05

Edit - works with multi matches too - will give a row per match, even if the same id has previously matched - you can then order by in_list or ids as appropriate.

This no longer conforms to the desired output since the question was edited, but I will leave it here as it's simple and contains the same information.