Setting up a linear optimizer with an “or” constraint

Question

I have a big linear optimizer I'm running and need some help setting up the constraints to get what I want. It's hard for me to express it in words exactly (hence the vague post-title), so I've written an example, details:

1. Select a total of 5 items, maximizing value and keeping the cost under 5k.
2. Each item has 2 "types". They are either labeled type1 = A, B, C, D, or E, and either type2 = X or Y.
3. 4 items must be type X, 1 must be type Y

The below example works great, but I want to add two more constraints and I'm not really sure how to do it. The two other constraints:

1. I want every optimization to have at least 2 instances of type1. I don't care which type there's multiple of or if two different types are multiples (e.g. 2 A's and 2 C's), which is why I'm thinking of it as an "or" constraint (A > 2 OR B > 2 OR...). Is this even doable?
2. This one might be a bit more difficult: Whichever "type Y" is chosen, I don't want that type1 to show up again. So say the Y item is type1 = C, I want every other chosen item to not be C. I imagine I'll need to add another dummy matrix interacting type1 and type2

Example of desired result:

   id type1 type2 value cost 
10 10     B     X    19  865
11 11     C     Y    19 1097 
18 18     D     X    19 1005
40 40     B     X    20  956
45 45     A     X    20  980

WORKING EXAMPLE:

library(lpSolve)
library(dplyr)

# setup df
  id <- 1:50
  type1 <- sample(c('A', 'B', 'C', 'D', 'E'), length(id), replace = T)
  type2 <- sample(c('X', 'X', 'X', 'Y'), length(id), replace = T)
  value <- round(runif(length(id), 0, 20),0)
  cost <- round(runif(length(id), 750, 1250),0)

  df <- data.frame(id, type1, type2, value, cost) %>% 
    mutate(total = 1)

# Attach dummy vars
  type1Dummy <- as.data.frame.matrix(table(df$id, df$type1))
  type2Dummy <- as.data.frame.matrix(table(df$id, df$type2))
  df <- cbind(df, type1Dummy, type2Dummy)

# constraints
  totalNum <- 5
  totalCost <- 5000
  totalX <- 4
  totalY <- 1
  rhs <- c(totalNum, totalCost, totalX, totalY)

# Direction vector
  numDir <- '=='
  costDir <- '<='
  xDir <- '=='
  yDir <- '=='
  dir <- c(numDir, costDir, xDir, yDir)

# Setup opt
  obj <- df$value
  mat <- t(data.frame(df$total, df$cost, df$X, df$Y))



# Solver Setup
  sol <- lpSolve::lp("max",   
                     objective.in = obj,
                     const.mat  = mat,
                     const.dir  = dir,
                     const.rhs  = rhs,
                     all.bin    = T
  )

  df$selected <- sol$solution
  dfSolved <- df[df$selected == 1,]
  dfSolved

Thank you for your help!