Step | Hyp | Ref
| Expression |
1 | | df-proj2 4568 |
. 2
⊢ Proj2 A = {x ∣ ( Phi x ∪
{0c}) ∈ A} |
2 | | vex 2862 |
. . . . . . 7
⊢ y ∈
V |
3 | | vex 2862 |
. . . . . . 7
⊢ x ∈
V |
4 | 2, 3 | opkelcnvk 4250 |
. . . . . 6
⊢ (⟪y, x⟫
∈ ◡k ∼ (( Ins2k Sk ⊕ Ins3k ((◡kImagek((Imagek((
Ins3k ∼ (( Ins3k Sk ∩
Ins2k Sk )
“k ℘1℘11c) ∖ ((
Ins2k Ins2k
Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k
SIk SIk
Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn
×k V)) ∪ ( Ik ∩ ( ∼ Nn
×k V))) ∘k Sk ) ∪ ({{0c}} ×k
V))) “k ℘1℘11c) ↔ ⟪x, y⟫ ∈ ∼ (( Ins2k Sk ⊕ Ins3k
((◡kImagek((Imagek((
Ins3k ∼ (( Ins3k Sk ∩
Ins2k Sk )
“k ℘1℘11c) ∖ ((
Ins2k Ins2k
Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k
SIk SIk
Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn
×k V)) ∪ ( Ik ∩ ( ∼ Nn
×k V))) ∘k Sk ) ∪ ({{0c}} ×k
V))) “k ℘1℘11c)) |
5 | | dfop2lem1 4573 |
. . . . . 6
⊢ (⟪x, y⟫
∈ ∼ (( Ins2k Sk ⊕ Ins3k ((◡kImagek((Imagek((
Ins3k ∼ (( Ins3k Sk ∩
Ins2k Sk )
“k ℘1℘11c) ∖ ((
Ins2k Ins2k
Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k
SIk SIk
Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn
×k V)) ∪ ( Ik ∩ ( ∼ Nn
×k V))) ∘k Sk ) ∪ ({{0c}} ×k
V))) “k ℘1℘11c) ↔ y = ( Phi x
∪ {0c})) |
6 | 4, 5 | bitri 240 |
. . . . 5
⊢ (⟪y, x⟫
∈ ◡k ∼ (( Ins2k Sk ⊕ Ins3k ((◡kImagek((Imagek((
Ins3k ∼ (( Ins3k Sk ∩
Ins2k Sk )
“k ℘1℘11c) ∖ ((
Ins2k Ins2k
Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k
SIk SIk
Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn
×k V)) ∪ ( Ik ∩ ( ∼ Nn
×k V))) ∘k Sk ) ∪ ({{0c}} ×k
V))) “k ℘1℘11c) ↔ y = ( Phi x
∪ {0c})) |
7 | 6 | rexbii 2639 |
. . . 4
⊢ (∃y ∈ A
⟪y, x⟫ ∈ ◡k ∼ (( Ins2k Sk ⊕ Ins3k ((◡kImagek((Imagek((
Ins3k ∼ (( Ins3k Sk ∩
Ins2k Sk )
“k ℘1℘11c) ∖ ((
Ins2k Ins2k
Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k
SIk SIk
Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn
×k V)) ∪ ( Ik ∩ ( ∼ Nn
×k V))) ∘k Sk ) ∪ ({{0c}} ×k
V))) “k ℘1℘11c) ↔ ∃y ∈
A y = ( Phi
x ∪ {0c})) |
8 | 3 | elimak 4259 |
. . . 4
⊢ (x ∈ (◡k ∼ (( Ins2k Sk ⊕ Ins3k ((◡kImagek((Imagek((
Ins3k ∼ (( Ins3k Sk ∩
Ins2k Sk )
“k ℘1℘11c) ∖ ((
Ins2k Ins2k
Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k
SIk SIk
Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn
×k V)) ∪ ( Ik ∩ ( ∼ Nn
×k V))) ∘k Sk ) ∪ ({{0c}} ×k
V))) “k ℘1℘11c) “k A) ↔ ∃y ∈ A
⟪y, x⟫ ∈ ◡k ∼ (( Ins2k Sk ⊕
Ins3k ((◡kImagek((Imagek((
Ins3k ∼ (( Ins3k Sk ∩
Ins2k Sk )
“k ℘1℘11c) ∖ ((
Ins2k Ins2k
Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k
SIk SIk
Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn
×k V)) ∪ ( Ik ∩ ( ∼ Nn
×k V))) ∘k Sk ) ∪ ({{0c}} ×k
V))) “k ℘1℘11c)) |
9 | | risset 2661 |
. . . 4
⊢ (( Phi x ∪
{0c}) ∈ A ↔ ∃y ∈ A y = ( Phi x ∪ {0c})) |
10 | 7, 8, 9 | 3bitr4i 268 |
. . 3
⊢ (x ∈ (◡k ∼ (( Ins2k Sk ⊕ Ins3k ((◡kImagek((Imagek((
Ins3k ∼ (( Ins3k Sk ∩
Ins2k Sk )
“k ℘1℘11c) ∖ ((
Ins2k Ins2k
Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k
SIk SIk
Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn
×k V)) ∪ ( Ik ∩ ( ∼ Nn
×k V))) ∘k Sk ) ∪ ({{0c}} ×k
V))) “k ℘1℘11c) “k A) ↔ ( Phi x ∪ {0c}) ∈ A) |
11 | 10 | abbi2i 2464 |
. 2
⊢ (◡k ∼ (( Ins2k Sk ⊕ Ins3k ((◡kImagek((Imagek((
Ins3k ∼ (( Ins3k Sk ∩
Ins2k Sk )
“k ℘1℘11c) ∖ ((
Ins2k Ins2k
Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k
SIk SIk
Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn
×k V)) ∪ ( Ik ∩ ( ∼ Nn
×k V))) ∘k Sk ) ∪ ({{0c}} ×k
V))) “k ℘1℘11c) “k A) = {x ∣ ( Phi x
∪ {0c}) ∈ A} |
12 | 1, 11 | eqtr4i 2376 |
1
⊢ Proj2 A = (◡k ∼ (( Ins2k Sk ⊕ Ins3k ((◡kImagek((Imagek((
Ins3k ∼ (( Ins3k Sk ∩
Ins2k Sk )
“k ℘1℘11c) ∖ ((
Ins2k Ins2k
Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k
SIk SIk
Sk )) “k ℘1℘1℘1℘11c)) “k ℘1℘11c) ∩ ( Nn
×k V)) ∪ ( Ik ∩ ( ∼ Nn
×k V))) ∘k Sk ) ∪ ({{0c}} ×k
V))) “k ℘1℘11c) “k A) |