Theorem relexprnd

Description: The range 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	relexprnd.1	|- ( ph -> N e. NN0 )
Assertion	relexprnd	|- ( ph -> ran ( R ^r N ) C_ U. U. R )

Proof

Step	Hyp	Ref	Expression
1		relexprnd.1	|- ( ph -> N e. NN0 )
2		relexprn	|- ( ( N e. NN0 /\ R e. _V ) -> ran ( R ^r N ) C_ U. U. R )
3	1 2	sylan	|- ( ( ph /\ R e. _V ) -> ran ( R ^r N ) C_ U. U. R )
4	3	ex	|- ( ph -> ( R e. _V -> ran ( R ^r N ) C_ U. U. R ) )
5		reldmrelexp	|- Rel dom ^r
6	5	ovprc1	|- ( -. R e. _V -> ( R ^r N ) = (/) )
7	6	rneqd	|- ( -. R e. _V -> ran ( R ^r N ) = ran (/) )
8		rn0	|- ran (/) = (/)
9	7 8	eqtrdi	|- ( -. R e. _V -> ran ( R ^r N ) = (/) )
10		0ss	|- (/) C_ U. U. R
11	9 10	eqsstrdi	|- ( -. R e. _V -> ran ( R ^r N ) C_ U. U. R )
12	4 11	pm2.61d1	|- ( ph -> ran ( R ^r N ) C_ U. U. R )