Comparing elements for equality

Suppose I have a dataframe as follows:

dat <- data.frame(id = c(1,1,2,2,2), var1 = c(10,10,20,20,25), var2 =
c('foo', 'foo', 'foo', 'foobar', 'foo'))

Now, if I were to subset by id, such as:

subset(dat, id==1)

  id var1 var2
1  1   10  foo
2  1   10  foo

I can see that the elements in var1 are exactly the same and the
elements in var2 are exactly the same. However,

subset(dat, id==2)

  id var1   var2
3  2   20    foo
4  2   20 foobar
5  2   25    foo

Shows the elements are not the same for either variable in this
instance. So, what I am looking to create is a data frame that would be
like this

id	freq	var1	var2
1	2	TRUE	TRUE	
2	3	FALSE	FALSE

Where freq is the number of times the ID is repeated in the dataframe. A
TRUE appears in the cell if all elements in the column are the same for
the ID and FALSE otherwise. It is insignificant which values differ for
my problem.

The way I am thinking about tackling this is to loop through the ID
variable and compare the values in the various columns of the dataframe.
The problem I am encountering is that I don't think all.equal or
identical are the right functions in this case.

So, say I was wanting to compare the elements of var1 for id ==1. I
would have

x <- c(10,10)

Of course, the following works

all.equal(x[1], x[2])

[1] TRUE

As would a similar call to identical. However, what if I only have a
vector of values (or if the column consists of names) that I want to
assess for equality when I am trying to automate a process over
thousands of cases? As in the example above, the vector may contain only
two values or it may contain many more. The number of values in the
vector differ by id.

Any thoughts?

Harold