Description: Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017)
Ref | Expression | ||
---|---|---|---|
Hypotheses | oppcbas.1 | ⊢ 𝑂 = ( oppCat ‘ 𝐶 ) | |
oppcid.2 | ⊢ 𝐵 = ( Id ‘ 𝐶 ) | ||
Assertion | oppcid | ⊢ ( 𝐶 ∈ Cat → ( Id ‘ 𝑂 ) = 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppcbas.1 | ⊢ 𝑂 = ( oppCat ‘ 𝐶 ) | |
2 | oppcid.2 | ⊢ 𝐵 = ( Id ‘ 𝐶 ) | |
3 | 1 | oppccatid | ⊢ ( 𝐶 ∈ Cat → ( 𝑂 ∈ Cat ∧ ( Id ‘ 𝑂 ) = ( Id ‘ 𝐶 ) ) ) |
4 | 3 | simprd | ⊢ ( 𝐶 ∈ Cat → ( Id ‘ 𝑂 ) = ( Id ‘ 𝐶 ) ) |
5 | 4 2 | eqtr4di | ⊢ ( 𝐶 ∈ Cat → ( Id ‘ 𝑂 ) = 𝐵 ) |