Step |
Hyp |
Ref |
Expression |
1 |
|
subcss1.1 |
⊢ ( 𝜑 → 𝐽 ∈ ( Subcat ‘ 𝐶 ) ) |
2 |
|
subcss1.2 |
⊢ ( 𝜑 → 𝐽 Fn ( 𝑆 × 𝑆 ) ) |
3 |
|
subcss2.h |
⊢ 𝐻 = ( Hom ‘ 𝐶 ) |
4 |
|
subcss2.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑆 ) |
5 |
|
subcss2.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑆 ) |
6 |
|
eqid |
⊢ ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝐶 ) |
7 |
1 6
|
subcssc |
⊢ ( 𝜑 → 𝐽 ⊆cat ( Homf ‘ 𝐶 ) ) |
8 |
2 7 4 5
|
ssc2 |
⊢ ( 𝜑 → ( 𝑋 𝐽 𝑌 ) ⊆ ( 𝑋 ( Homf ‘ 𝐶 ) 𝑌 ) ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
10 |
1 2 9
|
subcss1 |
⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝐶 ) ) |
11 |
10 4
|
sseldd |
⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐶 ) ) |
12 |
10 5
|
sseldd |
⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐶 ) ) |
13 |
6 9 3 11 12
|
homfval |
⊢ ( 𝜑 → ( 𝑋 ( Homf ‘ 𝐶 ) 𝑌 ) = ( 𝑋 𝐻 𝑌 ) ) |
14 |
8 13
|
sseqtrd |
⊢ ( 𝜑 → ( 𝑋 𝐽 𝑌 ) ⊆ ( 𝑋 𝐻 𝑌 ) ) |