This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem elcnvrefrels2

Description: Element of the class of converse reflexive relations. (Contributed by Peter Mazsa, 25-Jul-2019)

Ref Expression
Assertion elcnvrefrels2 ( 𝑅 ∈ CnvRefRels ↔ ( 𝑅 ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ 𝑅 ∈ Rels ) )

Proof

Step Hyp Ref Expression
1 dfcnvrefrels2 ⊢ CnvRefRels = { 𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ ( dom 𝑟 × ran 𝑟 ) ) }
2 id ⊢ ( 𝑟 = 𝑅 → 𝑟 = 𝑅 )
3 dmeq ⊢ ( 𝑟 = 𝑅 → dom 𝑟 = dom 𝑅 )
4 rneq ⊢ ( 𝑟 = 𝑅 → ran 𝑟 = ran 𝑅 )
5 3 4 xpeq12d ⊢ ( 𝑟 = 𝑅 → ( dom 𝑟 × ran 𝑟 ) = ( dom 𝑅 × ran 𝑅 ) )
6 5 ineq2d ⊢ ( 𝑟 = 𝑅 → ( I ∩ ( dom 𝑟 × ran 𝑟 ) ) = ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) )
7 2 6 sseq12d ⊢ ( 𝑟 = 𝑅 → ( 𝑟 ⊆ ( I ∩ ( dom 𝑟 × ran 𝑟 ) ) ↔ 𝑅 ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ) )
8 1 7 rabeqel ⊢ ( 𝑅 ∈ CnvRefRels ↔ ( 𝑅 ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ 𝑅 ∈ Rels ) )