Metamath Proof Explorer


Theorem cosscnvssid4

Description: Equivalent expressions for the class of cosets by the converse of R to be a subset of the identity class. (Contributed by Peter Mazsa, 31-Aug-2021)

Ref Expression
Assertion cosscnvssid4 ( ≀ 𝑅 ⊆ I ↔ ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 )

Proof

Step Hyp Ref Expression
1 cossssid4 ( ≀ 𝑅 ⊆ I ↔ ∀ 𝑥 ∃* 𝑢 𝑥 𝑅 𝑢 )
2 brcnvg ( ( 𝑥 ∈ V ∧ 𝑢 ∈ V ) → ( 𝑥 𝑅 𝑢𝑢 𝑅 𝑥 ) )
3 2 el2v ( 𝑥 𝑅 𝑢𝑢 𝑅 𝑥 )
4 3 mobii ( ∃* 𝑢 𝑥 𝑅 𝑢 ↔ ∃* 𝑢 𝑢 𝑅 𝑥 )
5 4 albii ( ∀ 𝑥 ∃* 𝑢 𝑥 𝑅 𝑢 ↔ ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 )
6 1 5 bitri ( ≀ 𝑅 ⊆ I ↔ ∀ 𝑥 ∃* 𝑢 𝑢 𝑅 𝑥 )