Step |
Hyp |
Ref |
Expression |
1 |
|
coeq2 |
⊢ ( 𝑓 = 𝐹 → ( ℜ ∘ 𝑓 ) = ( ℜ ∘ 𝐹 ) ) |
2 |
1
|
cnveqd |
⊢ ( 𝑓 = 𝐹 → ◡ ( ℜ ∘ 𝑓 ) = ◡ ( ℜ ∘ 𝐹 ) ) |
3 |
2
|
imaeq1d |
⊢ ( 𝑓 = 𝐹 → ( ◡ ( ℜ ∘ 𝑓 ) “ 𝑥 ) = ( ◡ ( ℜ ∘ 𝐹 ) “ 𝑥 ) ) |
4 |
3
|
eleq1d |
⊢ ( 𝑓 = 𝐹 → ( ( ◡ ( ℜ ∘ 𝑓 ) “ 𝑥 ) ∈ dom vol ↔ ( ◡ ( ℜ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ) |
5 |
|
coeq2 |
⊢ ( 𝑓 = 𝐹 → ( ℑ ∘ 𝑓 ) = ( ℑ ∘ 𝐹 ) ) |
6 |
5
|
cnveqd |
⊢ ( 𝑓 = 𝐹 → ◡ ( ℑ ∘ 𝑓 ) = ◡ ( ℑ ∘ 𝐹 ) ) |
7 |
6
|
imaeq1d |
⊢ ( 𝑓 = 𝐹 → ( ◡ ( ℑ ∘ 𝑓 ) “ 𝑥 ) = ( ◡ ( ℑ ∘ 𝐹 ) “ 𝑥 ) ) |
8 |
7
|
eleq1d |
⊢ ( 𝑓 = 𝐹 → ( ( ◡ ( ℑ ∘ 𝑓 ) “ 𝑥 ) ∈ dom vol ↔ ( ◡ ( ℑ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ) |
9 |
4 8
|
anbi12d |
⊢ ( 𝑓 = 𝐹 → ( ( ( ◡ ( ℜ ∘ 𝑓 ) “ 𝑥 ) ∈ dom vol ∧ ( ◡ ( ℑ ∘ 𝑓 ) “ 𝑥 ) ∈ dom vol ) ↔ ( ( ◡ ( ℜ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ∧ ( ◡ ( ℑ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ) ) |
10 |
9
|
ralbidv |
⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑥 ∈ ran (,) ( ( ◡ ( ℜ ∘ 𝑓 ) “ 𝑥 ) ∈ dom vol ∧ ( ◡ ( ℑ ∘ 𝑓 ) “ 𝑥 ) ∈ dom vol ) ↔ ∀ 𝑥 ∈ ran (,) ( ( ◡ ( ℜ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ∧ ( ◡ ( ℑ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ) ) |
11 |
|
df-mbf |
⊢ MblFn = { 𝑓 ∈ ( ℂ ↑pm ℝ ) ∣ ∀ 𝑥 ∈ ran (,) ( ( ◡ ( ℜ ∘ 𝑓 ) “ 𝑥 ) ∈ dom vol ∧ ( ◡ ( ℑ ∘ 𝑓 ) “ 𝑥 ) ∈ dom vol ) } |
12 |
10 11
|
elrab2 |
⊢ ( 𝐹 ∈ MblFn ↔ ( 𝐹 ∈ ( ℂ ↑pm ℝ ) ∧ ∀ 𝑥 ∈ ran (,) ( ( ◡ ( ℜ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ∧ ( ◡ ( ℑ ∘ 𝐹 ) “ 𝑥 ) ∈ dom vol ) ) ) |