Theorem dfcnvrefrel2

Description: Alternate definition of the converse reflexive relation predicate. (Contributed by Peter Mazsa, 24-Jul-2019)

	Ref	Expression
Assertion	dfcnvrefrel2	⊢ ( CnvRefRel 𝑅 ↔ ( 𝑅 ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ 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		dfrel6	⊢ ( Rel 𝑅 ↔ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) = 𝑅 )
3	2	biimpi	⊢ ( Rel 𝑅 → ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) = 𝑅 )
4	3	sseq1d	⊢ ( Rel 𝑅 → ( ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ↔ 𝑅 ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ) )
5	4	pm5.32ri	⊢ ( ( ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 ) ↔ ( 𝑅 ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 ) )
6	1 5	bitri	⊢ ( CnvRefRel 𝑅 ↔ ( 𝑅 ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 ) )