Metamath Proof Explorer


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 ( 𝜑 → Rel 𝑅 )
fvrcllb0da.r ( 𝜑𝑅 ∈ V )
Assertion fvrcllb0da ( 𝜑 → ( I ↾ 𝑅 ) ⊆ ( r* ‘ 𝑅 ) )

Proof

Step Hyp Ref Expression
1 fvrcllb0da.rel ( 𝜑 → Rel 𝑅 )
2 fvrcllb0da.r ( 𝜑𝑅 ∈ V )
3 dfrcl4 r* = ( 𝑟 ∈ V ↦ 𝑛 ∈ { 0 , 1 } ( 𝑟𝑟 𝑛 ) )
4 prex { 0 , 1 } ∈ V
5 4 a1i ( 𝜑 → { 0 , 1 } ∈ V )
6 c0ex 0 ∈ V
7 6 prid1 0 ∈ { 0 , 1 }
8 7 a1i ( 𝜑 → 0 ∈ { 0 , 1 } )
9 3 2 5 1 8 fvmptiunrelexplb0da ( 𝜑 → ( I ↾ 𝑅 ) ⊆ ( r* ‘ 𝑅 ) )