Description: A composition is not affected by a double converse of its second argument. (Contributed by NM, 8-Oct-2007)
Ref | Expression | ||
---|---|---|---|
Assertion | cocnvcnv2 | ⊢ ( 𝐴 ∘ ◡ ◡ 𝐵 ) = ( 𝐴 ∘ 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvcnv2 | ⊢ ◡ ◡ 𝐵 = ( 𝐵 ↾ V ) | |
2 | 1 | coeq2i | ⊢ ( 𝐴 ∘ ◡ ◡ 𝐵 ) = ( 𝐴 ∘ ( 𝐵 ↾ V ) ) |
3 | resco | ⊢ ( ( 𝐴 ∘ 𝐵 ) ↾ V ) = ( 𝐴 ∘ ( 𝐵 ↾ V ) ) | |
4 | relco | ⊢ Rel ( 𝐴 ∘ 𝐵 ) | |
5 | dfrel3 | ⊢ ( Rel ( 𝐴 ∘ 𝐵 ) ↔ ( ( 𝐴 ∘ 𝐵 ) ↾ V ) = ( 𝐴 ∘ 𝐵 ) ) | |
6 | 4 5 | mpbi | ⊢ ( ( 𝐴 ∘ 𝐵 ) ↾ V ) = ( 𝐴 ∘ 𝐵 ) |
7 | 2 3 6 | 3eqtr2i | ⊢ ( 𝐴 ∘ ◡ ◡ 𝐵 ) = ( 𝐴 ∘ 𝐵 ) |