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Metamath Proof Explorer

Theorem zerooval

Description: The value of the zero object function, i.e. the set of all zero objects of a category. (Contributed by AV, 3-Apr-2020)

Ref Expression
Hypotheses initoval.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
initoval.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
initoval.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
Assertion zerooval ( 𝜑 → ( ZeroO ‘ 𝐶 ) = ( ( InitO ‘ 𝐶 ) ∩ ( TermO ‘ 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 initoval.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
2 initoval.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
3 initoval.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
4 df-zeroo ⊢ ZeroO = ( 𝑐 ∈ Cat ↦ ( ( InitO ‘ 𝑐 ) ∩ ( TermO ‘ 𝑐 ) ) )
5 fveq2 ⊢ ( 𝑐 = 𝐶 → ( InitO ‘ 𝑐 ) = ( InitO ‘ 𝐶 ) )
6 fveq2 ⊢ ( 𝑐 = 𝐶 → ( TermO ‘ 𝑐 ) = ( TermO ‘ 𝐶 ) )
7 5 6 ineq12d ⊢ ( 𝑐 = 𝐶 → ( ( InitO ‘ 𝑐 ) ∩ ( TermO ‘ 𝑐 ) ) = ( ( InitO ‘ 𝐶 ) ∩ ( TermO ‘ 𝐶 ) ) )
8 fvex ⊢ ( InitO ‘ 𝐶 ) ∈ V
9 8 inex1 ⊢ ( ( InitO ‘ 𝐶 ) ∩ ( TermO ‘ 𝐶 ) ) ∈ V
10 9 a1i ⊢ ( 𝜑 → ( ( InitO ‘ 𝐶 ) ∩ ( TermO ‘ 𝐶 ) ) ∈ V )
11 4 7 1 10 fvmptd3 ⊢ ( 𝜑 → ( ZeroO ‘ 𝐶 ) = ( ( InitO ‘ 𝐶 ) ∩ ( TermO ‘ 𝐶 ) ) )