Theorem fvrtrcllb0d

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

	Ref	Expression
Hypothesis	fvrtrcllb0d.r	|- ( ph -> R e. _V )
Assertion	fvrtrcllb0d	|- ( ph -> ( _I |` ( dom R u. ran R ) ) C_ ( t* ` R ) )

Proof

Step	Hyp	Ref	Expression
1		fvrtrcllb0d.r	|- ( ph -> R e. _V )
2		dfrtrcl3	|- t* = ( r e. _V |-> U_ n e. NN0 ( r ^r n ) )
3		nn0ex	|- NN0 e. _V
4	3	a1i	|- ( ph -> NN0 e. _V )
5		0nn0	|- 0 e. NN0
6	5	a1i	|- ( ph -> 0 e. NN0 )
7	2 1 4 6	fvmptiunrelexplb0d	|- ( ph -> ( _I |` ( dom R u. ran R ) ) C_ ( t* ` R ) )