Step |
Hyp |
Ref |
Expression |
1 |
|
ulmpm |
⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝐹 ∈ ( ( ℂ ↑m 𝑆 ) ↑pm ℤ ) ) |
2 |
|
ovex |
⊢ ( ℂ ↑m 𝑆 ) ∈ V |
3 |
|
zex |
⊢ ℤ ∈ V |
4 |
2 3
|
elpm2 |
⊢ ( 𝐹 ∈ ( ( ℂ ↑m 𝑆 ) ↑pm ℤ ) ↔ ( 𝐹 : dom 𝐹 ⟶ ( ℂ ↑m 𝑆 ) ∧ dom 𝐹 ⊆ ℤ ) ) |
5 |
4
|
simplbi |
⊢ ( 𝐹 ∈ ( ( ℂ ↑m 𝑆 ) ↑pm ℤ ) → 𝐹 : dom 𝐹 ⟶ ( ℂ ↑m 𝑆 ) ) |
6 |
1 5
|
syl |
⊢ ( 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 → 𝐹 : dom 𝐹 ⟶ ( ℂ ↑m 𝑆 ) ) |
7 |
6
|
adantl |
⊢ ( ( 𝐹 Fn 𝑍 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ) → 𝐹 : dom 𝐹 ⟶ ( ℂ ↑m 𝑆 ) ) |
8 |
|
fndm |
⊢ ( 𝐹 Fn 𝑍 → dom 𝐹 = 𝑍 ) |
9 |
8
|
adantr |
⊢ ( ( 𝐹 Fn 𝑍 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ) → dom 𝐹 = 𝑍 ) |
10 |
9
|
feq2d |
⊢ ( ( 𝐹 Fn 𝑍 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ) → ( 𝐹 : dom 𝐹 ⟶ ( ℂ ↑m 𝑆 ) ↔ 𝐹 : 𝑍 ⟶ ( ℂ ↑m 𝑆 ) ) ) |
11 |
7 10
|
mpbid |
⊢ ( ( 𝐹 Fn 𝑍 ∧ 𝐹 ( ⇝𝑢 ‘ 𝑆 ) 𝐺 ) → 𝐹 : 𝑍 ⟶ ( ℂ ↑m 𝑆 ) ) |