Metamath Proof Explorer


Theorem relexpdmd

Description: The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015) (Revised by RP, 30-May-2020) (Revised by AV, 12-Jul-2024)

Ref Expression
Hypothesis relexpdmd.1
|- ( ph -> N e. NN0 )
Assertion relexpdmd
|- ( ph -> dom ( R ^r N ) C_ U. U. R )

Proof

Step Hyp Ref Expression
1 relexpdmd.1
 |-  ( ph -> N e. NN0 )
2 relexpdm
 |-  ( ( N e. NN0 /\ R e. _V ) -> dom ( R ^r N ) C_ U. U. R )
3 1 2 sylan
 |-  ( ( ph /\ R e. _V ) -> dom ( R ^r N ) C_ U. U. R )
4 3 ex
 |-  ( ph -> ( R e. _V -> dom ( R ^r N ) C_ U. U. R ) )
5 reldmrelexp
 |-  Rel dom ^r
6 5 ovprc1
 |-  ( -. R e. _V -> ( R ^r N ) = (/) )
7 6 dmeqd
 |-  ( -. R e. _V -> dom ( R ^r N ) = dom (/) )
8 dm0
 |-  dom (/) = (/)
9 7 8 eqtrdi
 |-  ( -. R e. _V -> dom ( R ^r N ) = (/) )
10 0ss
 |-  (/) C_ U. U. R
11 9 10 eqsstrdi
 |-  ( -. R e. _V -> dom ( R ^r N ) C_ U. U. R )
12 4 11 pm2.61d1
 |-  ( ph -> dom ( R ^r N ) C_ U. U. R )