Description: The double converse of the restriction of a class. (Contributed by NM, 3-Jun-2007)
Ref | Expression | ||
---|---|---|---|
Assertion | cnvcnvres | ⊢ ◡ ◡ ( 𝐴 ↾ 𝐵 ) = ( ◡ ◡ 𝐴 ↾ 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres | ⊢ Rel ( 𝐴 ↾ 𝐵 ) | |
2 | dfrel2 | ⊢ ( Rel ( 𝐴 ↾ 𝐵 ) ↔ ◡ ◡ ( 𝐴 ↾ 𝐵 ) = ( 𝐴 ↾ 𝐵 ) ) | |
3 | 1 2 | mpbi | ⊢ ◡ ◡ ( 𝐴 ↾ 𝐵 ) = ( 𝐴 ↾ 𝐵 ) |
4 | rescnvcnv | ⊢ ( ◡ ◡ 𝐴 ↾ 𝐵 ) = ( 𝐴 ↾ 𝐵 ) | |
5 | 3 4 | eqtr4i | ⊢ ◡ ◡ ( 𝐴 ↾ 𝐵 ) = ( ◡ ◡ 𝐴 ↾ 𝐵 ) |