Description: Elementhood in a converse R -coset when R is a relation. (Contributed by Peter Mazsa, 9-Dec-2018)
Ref | Expression | ||
---|---|---|---|
Assertion | releleccnv | ⊢ ( Rel 𝑅 → ( 𝐴 ∈ [ 𝐵 ] ◡ 𝑅 ↔ 𝐴 𝑅 𝐵 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv | ⊢ Rel ◡ 𝑅 | |
2 | relelec | ⊢ ( Rel ◡ 𝑅 → ( 𝐴 ∈ [ 𝐵 ] ◡ 𝑅 ↔ 𝐵 ◡ 𝑅 𝐴 ) ) | |
3 | 1 2 | ax-mp | ⊢ ( 𝐴 ∈ [ 𝐵 ] ◡ 𝑅 ↔ 𝐵 ◡ 𝑅 𝐴 ) |
4 | relbrcnvg | ⊢ ( Rel 𝑅 → ( 𝐵 ◡ 𝑅 𝐴 ↔ 𝐴 𝑅 𝐵 ) ) | |
5 | 3 4 | syl5bb | ⊢ ( Rel 𝑅 → ( 𝐴 ∈ [ 𝐵 ] ◡ 𝑅 ↔ 𝐴 𝑅 𝐵 ) ) |