Theorem fvrcllb0da

Description: A restriction of the identity relation is a subset of the reflexive closure of a relation. (Contributed by RP, 22-Jul-2020)

	Ref	Expression
Hypotheses	fvrcllb0da.rel	|- ( ph -> Rel R )
	fvrcllb0da.r	|- ( ph -> R e. _V )
Assertion	fvrcllb0da	|- ( ph -> ( _I |` U. U. R ) C_ ( r* ` R ) )

Proof

Step	Hyp	Ref	Expression
1		fvrcllb0da.rel	|- ( ph -> Rel R )
2		fvrcllb0da.r	|- ( ph -> R e. _V )
3		dfrcl4	|- r* = ( r e. _V |-> U_ n e. { 0 , 1 } ( r ^r n ) )
4		prex	|- { 0 , 1 } e. _V
5	4	a1i	|- ( ph -> { 0 , 1 } e. _V )
6		c0ex	|- 0 e. _V
7	6	prid1	|- 0 e. { 0 , 1 }
8	7	a1i	|- ( ph -> 0 e. { 0 , 1 } )
9	3 2 5 1 8	fvmptiunrelexplb0da	|- ( ph -> ( _I |` U. U. R ) C_ ( r* ` R ) )