Theorem dfcnvrefrel4

Description: Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 25-May-2024)

	Ref	Expression
Assertion	dfcnvrefrel4	⊢ ( CnvRefRel 𝑅 ↔ ( 𝑅 ⊆ I ∧ Rel 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		df-cnvrefrel	⊢ ( CnvRefRel 𝑅 ↔ ( ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 ) )
2		cnvref4	⊢ ( Rel 𝑅 → ( ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ↔ 𝑅 ⊆ I ) )
3	1 2	bianim	⊢ ( CnvRefRel 𝑅 ↔ ( 𝑅 ⊆ I ∧ Rel 𝑅 ) )