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 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossssid4 | ⊢ ( ≀ ◡ 𝑅 ⊆ I ↔ ∀ 𝑥 ∃* 𝑢 𝑥 ◡ 𝑅 𝑢 ) | |
2 | brcnvg | ⊢ ( ( 𝑥 ∈ V ∧ 𝑢 ∈ V ) → ( 𝑥 ◡ 𝑅 𝑢 ↔ 𝑢 𝑅 𝑥 ) ) | |
3 | 2 | el2v | ⊢ ( 𝑥 ◡ 𝑅 𝑢 ↔ 𝑢 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 ) |