Step |
Hyp |
Ref |
Expression |
1 |
|
mclsval.d |
⊢ 𝐷 = ( mDV ‘ 𝑇 ) |
2 |
|
mclsval.e |
⊢ 𝐸 = ( mEx ‘ 𝑇 ) |
3 |
|
mclsval.c |
⊢ 𝐶 = ( mCls ‘ 𝑇 ) |
4 |
|
mclsval.1 |
⊢ ( 𝜑 → 𝑇 ∈ mFS ) |
5 |
|
mclsval.2 |
⊢ ( 𝜑 → 𝐾 ⊆ 𝐷 ) |
6 |
|
mclsval.3 |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐸 ) |
7 |
|
ss2mcls.4 |
⊢ ( 𝜑 → 𝑋 ⊆ 𝐾 ) |
8 |
|
ss2mcls.5 |
⊢ ( 𝜑 → 𝑌 ⊆ 𝐵 ) |
9 |
|
unss1 |
⊢ ( 𝑌 ⊆ 𝐵 → ( 𝑌 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ ( 𝐵 ∪ ran ( mVH ‘ 𝑇 ) ) ) |
10 |
|
sstr2 |
⊢ ( ( 𝑌 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ ( 𝐵 ∪ ran ( mVH ‘ 𝑇 ) ) → ( ( 𝐵 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 → ( 𝑌 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 ) ) |
11 |
8 9 10
|
3syl |
⊢ ( 𝜑 → ( ( 𝐵 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 → ( 𝑌 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 ) ) |
12 |
|
sstr2 |
⊢ ( ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 → ( 𝑋 ⊆ 𝐾 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) |
13 |
7 12
|
syl5com |
⊢ ( 𝜑 → ( ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) |
14 |
13
|
imim2d |
⊢ ( 𝜑 → ( ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 ) → ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) ) |
15 |
14
|
2alimdv |
⊢ ( 𝜑 → ( ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 ) → ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) ) |
16 |
15
|
anim2d |
⊢ ( 𝜑 → ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 ) ) → ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) ) ) |
17 |
16
|
imim1d |
⊢ ( 𝜑 → ( ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) → ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) |
18 |
17
|
ralimdv |
⊢ ( 𝜑 → ( ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) |
19 |
18
|
imim2d |
⊢ ( 𝜑 → ( ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) → ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) ) |
20 |
19
|
alimdv |
⊢ ( 𝜑 → ( ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) → ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) ) |
21 |
20
|
2alimdv |
⊢ ( 𝜑 → ( ∀ 𝑚 ∀ 𝑜 ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) → ∀ 𝑚 ∀ 𝑜 ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) ) |
22 |
11 21
|
anim12d |
⊢ ( 𝜑 → ( ( ( 𝐵 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 ∧ ∀ 𝑚 ∀ 𝑜 ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) → ( ( 𝑌 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 ∧ ∀ 𝑚 ∀ 𝑜 ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) ) ) |
23 |
22
|
ss2abdv |
⊢ ( 𝜑 → { 𝑐 ∣ ( ( 𝐵 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 ∧ ∀ 𝑚 ∀ 𝑜 ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) } ⊆ { 𝑐 ∣ ( ( 𝑌 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 ∧ ∀ 𝑚 ∀ 𝑜 ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) } ) |
24 |
|
intss |
⊢ ( { 𝑐 ∣ ( ( 𝐵 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 ∧ ∀ 𝑚 ∀ 𝑜 ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) } ⊆ { 𝑐 ∣ ( ( 𝑌 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 ∧ ∀ 𝑚 ∀ 𝑜 ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) } → ∩ { 𝑐 ∣ ( ( 𝑌 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 ∧ ∀ 𝑚 ∀ 𝑜 ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) } ⊆ ∩ { 𝑐 ∣ ( ( 𝐵 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 ∧ ∀ 𝑚 ∀ 𝑜 ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) } ) |
25 |
23 24
|
syl |
⊢ ( 𝜑 → ∩ { 𝑐 ∣ ( ( 𝑌 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 ∧ ∀ 𝑚 ∀ 𝑜 ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) } ⊆ ∩ { 𝑐 ∣ ( ( 𝐵 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 ∧ ∀ 𝑚 ∀ 𝑜 ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) } ) |
26 |
7 5
|
sstrd |
⊢ ( 𝜑 → 𝑋 ⊆ 𝐷 ) |
27 |
8 6
|
sstrd |
⊢ ( 𝜑 → 𝑌 ⊆ 𝐸 ) |
28 |
|
eqid |
⊢ ( mVH ‘ 𝑇 ) = ( mVH ‘ 𝑇 ) |
29 |
|
eqid |
⊢ ( mAx ‘ 𝑇 ) = ( mAx ‘ 𝑇 ) |
30 |
|
eqid |
⊢ ( mSubst ‘ 𝑇 ) = ( mSubst ‘ 𝑇 ) |
31 |
|
eqid |
⊢ ( mVars ‘ 𝑇 ) = ( mVars ‘ 𝑇 ) |
32 |
1 2 3 4 26 27 28 29 30 31
|
mclsval |
⊢ ( 𝜑 → ( 𝑋 𝐶 𝑌 ) = ∩ { 𝑐 ∣ ( ( 𝑌 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 ∧ ∀ 𝑚 ∀ 𝑜 ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝑋 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) } ) |
33 |
1 2 3 4 5 6 28 29 30 31
|
mclsval |
⊢ ( 𝜑 → ( 𝐾 𝐶 𝐵 ) = ∩ { 𝑐 ∣ ( ( 𝐵 ∪ ran ( mVH ‘ 𝑇 ) ) ⊆ 𝑐 ∧ ∀ 𝑚 ∀ 𝑜 ∀ 𝑝 ( 〈 𝑚 , 𝑜 , 𝑝 〉 ∈ ( mAx ‘ 𝑇 ) → ∀ 𝑠 ∈ ran ( mSubst ‘ 𝑇 ) ( ( ( 𝑠 “ ( 𝑜 ∪ ran ( mVH ‘ 𝑇 ) ) ) ⊆ 𝑐 ∧ ∀ 𝑥 ∀ 𝑦 ( 𝑥 𝑚 𝑦 → ( ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑥 ) ) ) × ( ( mVars ‘ 𝑇 ) ‘ ( 𝑠 ‘ ( ( mVH ‘ 𝑇 ) ‘ 𝑦 ) ) ) ) ⊆ 𝐾 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) } ) |
34 |
25 32 33
|
3sstr4d |
⊢ ( 𝜑 → ( 𝑋 𝐶 𝑌 ) ⊆ ( 𝐾 𝐶 𝐵 ) ) |