Theorem fvrtrcllb0da

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

	Ref	Expression
Hypotheses	fvrtrcllb0da.rel	|- ( ph -> Rel R )
	fvrtrcllb0da.r	|- ( ph -> R e. _V )
Assertion	fvrtrcllb0da	|- ( ph -> ( _I |` U. U. R ) C_ ( t* ` R ) )

Proof

Step	Hyp	Ref	Expression
1		fvrtrcllb0da.rel	|- ( ph -> Rel R )
2		fvrtrcllb0da.r	|- ( ph -> R e. _V )
3		dfrtrcl3	|- t* = ( r e. _V |-> U_ n e. NN0 ( r ^r n ) )
4		nn0ex	|- NN0 e. _V
5	4	a1i	|- ( ph -> NN0 e. _V )
6		0nn0	|- 0 e. NN0
7	6	a1i	|- ( ph -> 0 e. NN0 )
8	3 2 5 1 7	fvmptiunrelexplb0da	|- ( ph -> ( _I |` U. U. R ) C_ ( t* ` R ) )