Step |
Hyp |
Ref |
Expression |
1 |
|
resvlem.r |
⊢ 𝑅 = ( 𝑊 ↾v 𝐴 ) |
2 |
|
resvlem.e |
⊢ 𝐶 = ( 𝐸 ‘ 𝑊 ) |
3 |
|
resvlem.f |
⊢ 𝐸 = Slot ( 𝐸 ‘ ndx ) |
4 |
|
resvlem.n |
⊢ ( 𝐸 ‘ ndx ) ≠ ( Scalar ‘ ndx ) |
5 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
6 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) |
7 |
1 5 6
|
resvid2 |
⊢ ( ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → 𝑅 = 𝑊 ) |
8 |
7
|
fveq2d |
⊢ ( ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐸 ‘ 𝑅 ) = ( 𝐸 ‘ 𝑊 ) ) |
9 |
8
|
3expib |
⊢ ( ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ 𝐴 → ( ( 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐸 ‘ 𝑅 ) = ( 𝐸 ‘ 𝑊 ) ) ) |
10 |
1 5 6
|
resvval2 |
⊢ ( ( ¬ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → 𝑅 = ( 𝑊 sSet 〈 ( Scalar ‘ ndx ) , ( ( Scalar ‘ 𝑊 ) ↾s 𝐴 ) 〉 ) ) |
11 |
10
|
fveq2d |
⊢ ( ( ¬ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐸 ‘ 𝑅 ) = ( 𝐸 ‘ ( 𝑊 sSet 〈 ( Scalar ‘ ndx ) , ( ( Scalar ‘ 𝑊 ) ↾s 𝐴 ) 〉 ) ) ) |
12 |
3 4
|
setsnid |
⊢ ( 𝐸 ‘ 𝑊 ) = ( 𝐸 ‘ ( 𝑊 sSet 〈 ( Scalar ‘ ndx ) , ( ( Scalar ‘ 𝑊 ) ↾s 𝐴 ) 〉 ) ) |
13 |
11 12
|
eqtr4di |
⊢ ( ( ¬ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐸 ‘ 𝑅 ) = ( 𝐸 ‘ 𝑊 ) ) |
14 |
13
|
3expib |
⊢ ( ¬ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ⊆ 𝐴 → ( ( 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐸 ‘ 𝑅 ) = ( 𝐸 ‘ 𝑊 ) ) ) |
15 |
9 14
|
pm2.61i |
⊢ ( ( 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐸 ‘ 𝑅 ) = ( 𝐸 ‘ 𝑊 ) ) |
16 |
3
|
str0 |
⊢ ∅ = ( 𝐸 ‘ ∅ ) |
17 |
16
|
eqcomi |
⊢ ( 𝐸 ‘ ∅ ) = ∅ |
18 |
|
reldmresv |
⊢ Rel dom ↾v |
19 |
17 1 18
|
oveqprc |
⊢ ( ¬ 𝑊 ∈ V → ( 𝐸 ‘ 𝑊 ) = ( 𝐸 ‘ 𝑅 ) ) |
20 |
19
|
eqcomd |
⊢ ( ¬ 𝑊 ∈ V → ( 𝐸 ‘ 𝑅 ) = ( 𝐸 ‘ 𝑊 ) ) |
21 |
20
|
adantr |
⊢ ( ( ¬ 𝑊 ∈ V ∧ 𝐴 ∈ 𝑉 ) → ( 𝐸 ‘ 𝑅 ) = ( 𝐸 ‘ 𝑊 ) ) |
22 |
15 21
|
pm2.61ian |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝐸 ‘ 𝑅 ) = ( 𝐸 ‘ 𝑊 ) ) |
23 |
2 22
|
eqtr4id |
⊢ ( 𝐴 ∈ 𝑉 → 𝐶 = ( 𝐸 ‘ 𝑅 ) ) |