Theorem fvrcllb0d

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

	Ref	Expression
Hypothesis	fvrcllb0d.r	|- ( ph -> R e. _V )
Assertion	fvrcllb0d	|- ( ph -> ( _I |` ( dom R u. ran R ) ) C_ ( r* ` R ) )

Proof

Step	Hyp	Ref	Expression
1		fvrcllb0d.r	|- ( ph -> R e. _V )
2		dfrcl4	|- r* = ( r e. _V |-> U_ n e. { 0 , 1 } ( r ^r n ) )
3		prex	|- { 0 , 1 } e. _V
4	3	a1i	|- ( ph -> { 0 , 1 } e. _V )
5		c0ex	|- 0 e. _V
6	5	prid1	|- 0 e. { 0 , 1 }
7	6	a1i	|- ( ph -> 0 e. { 0 , 1 } )
8	2 1 4 7	fvmptiunrelexplb0d	|- ( ph -> ( _I |` ( dom R u. ran R ) ) C_ ( r* ` R ) )