Theorem dfrel5

Description: Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 6-Nov-2018)

	Ref	Expression
Assertion	dfrel5	⊢ ( Rel 𝑅 ↔ ( 𝑅 ↾ dom 𝑅 ) = 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		dfrel2	⊢ ( Rel 𝑅 ↔ ◡ ◡ 𝑅 = 𝑅 )
2		resdm2	⊢ ( 𝑅 ↾ dom 𝑅 ) = ◡ ◡ 𝑅
3	2	eqeq1i	⊢ ( ( 𝑅 ↾ dom 𝑅 ) = 𝑅 ↔ ◡ ◡ 𝑅 = 𝑅 )
4	1 3	bitr4i	⊢ ( Rel 𝑅 ↔ ( 𝑅 ↾ dom 𝑅 ) = 𝑅 )