Metamath Proof Explorer
Description: Infer subset relation on objects from the subcategory subset relation.
(Contributed by Mario Carneiro, 6-Jan-2017)
|
|
Ref |
Expression |
|
Hypotheses |
isssc.1 |
⊢ ( 𝜑 → 𝐻 Fn ( 𝑆 × 𝑆 ) ) |
|
|
isssc.2 |
⊢ ( 𝜑 → 𝐽 Fn ( 𝑇 × 𝑇 ) ) |
|
|
ssc1.3 |
⊢ ( 𝜑 → 𝐻 ⊆cat 𝐽 ) |
|
Assertion |
ssc1 |
⊢ ( 𝜑 → 𝑆 ⊆ 𝑇 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
isssc.1 |
⊢ ( 𝜑 → 𝐻 Fn ( 𝑆 × 𝑆 ) ) |
2 |
|
isssc.2 |
⊢ ( 𝜑 → 𝐽 Fn ( 𝑇 × 𝑇 ) ) |
3 |
|
ssc1.3 |
⊢ ( 𝜑 → 𝐻 ⊆cat 𝐽 ) |
4 |
|
sscrel |
⊢ Rel ⊆cat |
5 |
4
|
brrelex2i |
⊢ ( 𝐻 ⊆cat 𝐽 → 𝐽 ∈ V ) |
6 |
3 5
|
syl |
⊢ ( 𝜑 → 𝐽 ∈ V ) |
7 |
2
|
ssclem |
⊢ ( 𝜑 → ( 𝐽 ∈ V ↔ 𝑇 ∈ V ) ) |
8 |
6 7
|
mpbid |
⊢ ( 𝜑 → 𝑇 ∈ V ) |
9 |
1 2 8
|
isssc |
⊢ ( 𝜑 → ( 𝐻 ⊆cat 𝐽 ↔ ( 𝑆 ⊆ 𝑇 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐻 𝑦 ) ⊆ ( 𝑥 𝐽 𝑦 ) ) ) ) |
10 |
3 9
|
mpbid |
⊢ ( 𝜑 → ( 𝑆 ⊆ 𝑇 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐻 𝑦 ) ⊆ ( 𝑥 𝐽 𝑦 ) ) ) |
11 |
10
|
simpld |
⊢ ( 𝜑 → 𝑆 ⊆ 𝑇 ) |