Step |
Hyp |
Ref |
Expression |
1 |
|
0ex |
⊢ ∅ ∈ V |
2 |
|
satefv |
⊢ ( ( ∅ ∈ V ∧ 𝑈 ∈ 𝑉 ) → ( ∅ Sat∈ 𝑈 ) = ( ( ( ∅ Sat ( E ∩ ( ∅ × ∅ ) ) ) ‘ ω ) ‘ 𝑈 ) ) |
3 |
1 2
|
mpan |
⊢ ( 𝑈 ∈ 𝑉 → ( ∅ Sat∈ 𝑈 ) = ( ( ( ∅ Sat ( E ∩ ( ∅ × ∅ ) ) ) ‘ ω ) ‘ 𝑈 ) ) |
4 |
|
xp0 |
⊢ ( ∅ × ∅ ) = ∅ |
5 |
4
|
ineq2i |
⊢ ( E ∩ ( ∅ × ∅ ) ) = ( E ∩ ∅ ) |
6 |
|
in0 |
⊢ ( E ∩ ∅ ) = ∅ |
7 |
5 6
|
eqtri |
⊢ ( E ∩ ( ∅ × ∅ ) ) = ∅ |
8 |
7
|
oveq2i |
⊢ ( ∅ Sat ( E ∩ ( ∅ × ∅ ) ) ) = ( ∅ Sat ∅ ) |
9 |
8
|
fveq1i |
⊢ ( ( ∅ Sat ( E ∩ ( ∅ × ∅ ) ) ) ‘ ω ) = ( ( ∅ Sat ∅ ) ‘ ω ) |
10 |
9
|
fveq1i |
⊢ ( ( ( ∅ Sat ( E ∩ ( ∅ × ∅ ) ) ) ‘ ω ) ‘ 𝑈 ) = ( ( ( ∅ Sat ∅ ) ‘ ω ) ‘ 𝑈 ) |
11 |
3 10
|
eqtrdi |
⊢ ( 𝑈 ∈ 𝑉 → ( ∅ Sat∈ 𝑈 ) = ( ( ( ∅ Sat ∅ ) ‘ ω ) ‘ 𝑈 ) ) |