Step |
Hyp |
Ref |
Expression |
1 |
|
mclsval.d |
⊢ 𝐷 = ( mDV ‘ 𝑇 ) |
2 |
|
mclsval.e |
⊢ 𝐸 = ( mEx ‘ 𝑇 ) |
3 |
|
mclsval.c |
⊢ 𝐶 = ( mCls ‘ 𝑇 ) |
4 |
|
n0i |
⊢ ( 𝐴 ∈ ( 𝐾 𝐶 𝐵 ) → ¬ ( 𝐾 𝐶 𝐵 ) = ∅ ) |
5 |
|
fvprc |
⊢ ( ¬ 𝑇 ∈ V → ( mCls ‘ 𝑇 ) = ∅ ) |
6 |
3 5
|
syl5eq |
⊢ ( ¬ 𝑇 ∈ V → 𝐶 = ∅ ) |
7 |
6
|
oveqd |
⊢ ( ¬ 𝑇 ∈ V → ( 𝐾 𝐶 𝐵 ) = ( 𝐾 ∅ 𝐵 ) ) |
8 |
|
0ov |
⊢ ( 𝐾 ∅ 𝐵 ) = ∅ |
9 |
7 8
|
eqtrdi |
⊢ ( ¬ 𝑇 ∈ V → ( 𝐾 𝐶 𝐵 ) = ∅ ) |
10 |
4 9
|
nsyl2 |
⊢ ( 𝐴 ∈ ( 𝐾 𝐶 𝐵 ) → 𝑇 ∈ V ) |
11 |
|
fveq2 |
⊢ ( 𝑡 = 𝑇 → ( mCls ‘ 𝑡 ) = ( mCls ‘ 𝑇 ) ) |
12 |
11 3
|
eqtr4di |
⊢ ( 𝑡 = 𝑇 → ( mCls ‘ 𝑡 ) = 𝐶 ) |
13 |
12
|
oveqd |
⊢ ( 𝑡 = 𝑇 → ( 𝐾 ( mCls ‘ 𝑡 ) 𝐵 ) = ( 𝐾 𝐶 𝐵 ) ) |
14 |
13
|
eleq2d |
⊢ ( 𝑡 = 𝑇 → ( 𝐴 ∈ ( 𝐾 ( mCls ‘ 𝑡 ) 𝐵 ) ↔ 𝐴 ∈ ( 𝐾 𝐶 𝐵 ) ) ) |
15 |
|
fvex |
⊢ ( mDV ‘ 𝑡 ) ∈ V |
16 |
15
|
elpw2 |
⊢ ( 𝐾 ∈ 𝒫 ( mDV ‘ 𝑡 ) ↔ 𝐾 ⊆ ( mDV ‘ 𝑡 ) ) |
17 |
|
fveq2 |
⊢ ( 𝑡 = 𝑇 → ( mDV ‘ 𝑡 ) = ( mDV ‘ 𝑇 ) ) |
18 |
17 1
|
eqtr4di |
⊢ ( 𝑡 = 𝑇 → ( mDV ‘ 𝑡 ) = 𝐷 ) |
19 |
18
|
sseq2d |
⊢ ( 𝑡 = 𝑇 → ( 𝐾 ⊆ ( mDV ‘ 𝑡 ) ↔ 𝐾 ⊆ 𝐷 ) ) |
20 |
16 19
|
syl5bb |
⊢ ( 𝑡 = 𝑇 → ( 𝐾 ∈ 𝒫 ( mDV ‘ 𝑡 ) ↔ 𝐾 ⊆ 𝐷 ) ) |
21 |
|
fvex |
⊢ ( mEx ‘ 𝑡 ) ∈ V |
22 |
21
|
elpw2 |
⊢ ( 𝐵 ∈ 𝒫 ( mEx ‘ 𝑡 ) ↔ 𝐵 ⊆ ( mEx ‘ 𝑡 ) ) |
23 |
|
fveq2 |
⊢ ( 𝑡 = 𝑇 → ( mEx ‘ 𝑡 ) = ( mEx ‘ 𝑇 ) ) |
24 |
23 2
|
eqtr4di |
⊢ ( 𝑡 = 𝑇 → ( mEx ‘ 𝑡 ) = 𝐸 ) |
25 |
24
|
sseq2d |
⊢ ( 𝑡 = 𝑇 → ( 𝐵 ⊆ ( mEx ‘ 𝑡 ) ↔ 𝐵 ⊆ 𝐸 ) ) |
26 |
22 25
|
syl5bb |
⊢ ( 𝑡 = 𝑇 → ( 𝐵 ∈ 𝒫 ( mEx ‘ 𝑡 ) ↔ 𝐵 ⊆ 𝐸 ) ) |
27 |
20 26
|
anbi12d |
⊢ ( 𝑡 = 𝑇 → ( ( 𝐾 ∈ 𝒫 ( mDV ‘ 𝑡 ) ∧ 𝐵 ∈ 𝒫 ( mEx ‘ 𝑡 ) ) ↔ ( 𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸 ) ) ) |
28 |
14 27
|
imbi12d |
⊢ ( 𝑡 = 𝑇 → ( ( 𝐴 ∈ ( 𝐾 ( mCls ‘ 𝑡 ) 𝐵 ) → ( 𝐾 ∈ 𝒫 ( mDV ‘ 𝑡 ) ∧ 𝐵 ∈ 𝒫 ( mEx ‘ 𝑡 ) ) ) ↔ ( 𝐴 ∈ ( 𝐾 𝐶 𝐵 ) → ( 𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸 ) ) ) ) |
29 |
|
vex |
⊢ 𝑡 ∈ V |
30 |
15
|
pwex |
⊢ 𝒫 ( mDV ‘ 𝑡 ) ∈ V |
31 |
21
|
pwex |
⊢ 𝒫 ( mEx ‘ 𝑡 ) ∈ V |
32 |
30 31
|
mpoex |
⊢ ( 𝑑 ∈ 𝒫 ( mDV ‘ 𝑡 ) , ℎ ∈ 𝒫 ( mEx ‘ 𝑡 ) ↦ ∩ { 𝑐 ∣ ( ( ℎ ∪ 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 ) ) ) ) ⊆ 𝑑 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) } ) ∈ V |
33 |
|
df-mcls |
⊢ mCls = ( 𝑡 ∈ V ↦ ( 𝑑 ∈ 𝒫 ( mDV ‘ 𝑡 ) , ℎ ∈ 𝒫 ( mEx ‘ 𝑡 ) ↦ ∩ { 𝑐 ∣ ( ( ℎ ∪ 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 ) ) ) ) ⊆ 𝑑 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) } ) ) |
34 |
33
|
fvmpt2 |
⊢ ( ( 𝑡 ∈ V ∧ ( 𝑑 ∈ 𝒫 ( mDV ‘ 𝑡 ) , ℎ ∈ 𝒫 ( mEx ‘ 𝑡 ) ↦ ∩ { 𝑐 ∣ ( ( ℎ ∪ 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 ) ) ) ) ⊆ 𝑑 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) } ) ∈ V ) → ( mCls ‘ 𝑡 ) = ( 𝑑 ∈ 𝒫 ( mDV ‘ 𝑡 ) , ℎ ∈ 𝒫 ( mEx ‘ 𝑡 ) ↦ ∩ { 𝑐 ∣ ( ( ℎ ∪ 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 ) ) ) ) ⊆ 𝑑 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) } ) ) |
35 |
29 32 34
|
mp2an |
⊢ ( mCls ‘ 𝑡 ) = ( 𝑑 ∈ 𝒫 ( mDV ‘ 𝑡 ) , ℎ ∈ 𝒫 ( mEx ‘ 𝑡 ) ↦ ∩ { 𝑐 ∣ ( ( ℎ ∪ 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 ) ) ) ) ⊆ 𝑑 ) ) → ( 𝑠 ‘ 𝑝 ) ∈ 𝑐 ) ) ) } ) |
36 |
35
|
elmpocl |
⊢ ( 𝐴 ∈ ( 𝐾 ( mCls ‘ 𝑡 ) 𝐵 ) → ( 𝐾 ∈ 𝒫 ( mDV ‘ 𝑡 ) ∧ 𝐵 ∈ 𝒫 ( mEx ‘ 𝑡 ) ) ) |
37 |
28 36
|
vtoclg |
⊢ ( 𝑇 ∈ V → ( 𝐴 ∈ ( 𝐾 𝐶 𝐵 ) → ( 𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸 ) ) ) |
38 |
10 37
|
mpcom |
⊢ ( 𝐴 ∈ ( 𝐾 𝐶 𝐵 ) → ( 𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸 ) ) |
39 |
38
|
simpld |
⊢ ( 𝐴 ∈ ( 𝐾 𝐶 𝐵 ) → 𝐾 ⊆ 𝐷 ) |
40 |
38
|
simprd |
⊢ ( 𝐴 ∈ ( 𝐾 𝐶 𝐵 ) → 𝐵 ⊆ 𝐸 ) |
41 |
10 39 40
|
3jca |
⊢ ( 𝐴 ∈ ( 𝐾 𝐶 𝐵 ) → ( 𝑇 ∈ V ∧ 𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸 ) ) |