Metamath Proof Explorer
Description: An element in the set of subcategories is a binary function.
(Contributed by Mario Carneiro, 4-Jan-2017)
|
|
Ref |
Expression |
|
Hypotheses |
subcixp.1 |
⊢ ( 𝜑 → 𝐽 ∈ ( Subcat ‘ 𝐶 ) ) |
|
|
subcfn.2 |
⊢ ( 𝜑 → 𝑆 = dom dom 𝐽 ) |
|
Assertion |
subcfn |
⊢ ( 𝜑 → 𝐽 Fn ( 𝑆 × 𝑆 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
subcixp.1 |
⊢ ( 𝜑 → 𝐽 ∈ ( Subcat ‘ 𝐶 ) ) |
2 |
|
subcfn.2 |
⊢ ( 𝜑 → 𝑆 = dom dom 𝐽 ) |
3 |
|
eqid |
⊢ ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐶 ) |
4 |
1 3
|
subcssc |
⊢ ( 𝜑 → 𝐽 ⊆cat ( Homf ‘ 𝐶 ) ) |
5 |
4 2
|
sscfn1 |
⊢ ( 𝜑 → 𝐽 Fn ( 𝑆 × 𝑆 ) ) |