Step |
Hyp |
Ref |
Expression |
1 |
|
mbfeqa.1 |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ ) |
2 |
|
mbfeqa.2 |
⊢ ( 𝜑 → ( vol* ‘ 𝐴 ) = 0 ) |
3 |
|
mbfeqa.3 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐵 ∖ 𝐴 ) ) → 𝐶 = 𝐷 ) |
4 |
|
mbfeqalem.4 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐶 ∈ ℝ ) |
5 |
|
mbfeqalem.5 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐷 ∈ ℝ ) |
6 |
|
inundif |
⊢ ( ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ∪ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ) = ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) |
7 |
|
incom |
⊢ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) = ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) |
8 |
|
dfin4 |
⊢ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) = ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∖ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ) |
9 |
7 8
|
eqtri |
⊢ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) = ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∖ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ) |
10 |
|
id |
⊢ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∈ dom vol → ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∈ dom vol ) |
11 |
1 2 3 4 5
|
mbfeqalem1 |
⊢ ( 𝜑 → ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ∈ dom vol ) |
12 |
|
difmbl |
⊢ ( ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∈ dom vol ∧ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ∈ dom vol ) → ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∖ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ) ∈ dom vol ) |
13 |
10 11 12
|
syl2anr |
⊢ ( ( 𝜑 ∧ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∈ dom vol ) → ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∖ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ) ∈ dom vol ) |
14 |
9 13
|
eqeltrid |
⊢ ( ( 𝜑 ∧ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∈ dom vol ) → ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ∈ dom vol ) |
15 |
3
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐵 ∖ 𝐴 ) ) → 𝐷 = 𝐶 ) |
16 |
1 2 15 5 4
|
mbfeqalem1 |
⊢ ( 𝜑 → ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ∈ dom vol ) |
17 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∈ dom vol ) → ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ∈ dom vol ) |
18 |
|
unmbl |
⊢ ( ( ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ∈ dom vol ∧ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ∈ dom vol ) → ( ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ∪ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ) ∈ dom vol ) |
19 |
14 17 18
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∈ dom vol ) → ( ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ∪ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ) ∈ dom vol ) |
20 |
6 19
|
eqeltrrid |
⊢ ( ( 𝜑 ∧ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∈ dom vol ) → ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∈ dom vol ) |
21 |
|
inundif |
⊢ ( ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ∪ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ) = ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) |
22 |
|
incom |
⊢ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) = ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) |
23 |
|
dfin4 |
⊢ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) = ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∖ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ) |
24 |
22 23
|
eqtri |
⊢ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) = ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∖ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ) |
25 |
|
id |
⊢ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∈ dom vol → ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∈ dom vol ) |
26 |
|
difmbl |
⊢ ( ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∈ dom vol ∧ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ∈ dom vol ) → ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∖ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ) ∈ dom vol ) |
27 |
25 16 26
|
syl2anr |
⊢ ( ( 𝜑 ∧ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∈ dom vol ) → ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∖ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ) ) ∈ dom vol ) |
28 |
24 27
|
eqeltrid |
⊢ ( ( 𝜑 ∧ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∈ dom vol ) → ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ∈ dom vol ) |
29 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∈ dom vol ) → ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ∈ dom vol ) |
30 |
|
unmbl |
⊢ ( ( ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ∈ dom vol ∧ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ∈ dom vol ) → ( ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ∪ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ) ∈ dom vol ) |
31 |
28 29 30
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∈ dom vol ) → ( ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∩ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ∪ ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∖ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ) ) ∈ dom vol ) |
32 |
21 31
|
eqeltrrid |
⊢ ( ( 𝜑 ∧ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∈ dom vol ) → ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∈ dom vol ) |
33 |
20 32
|
impbida |
⊢ ( 𝜑 → ( ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∈ dom vol ↔ ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∈ dom vol ) ) |
34 |
33
|
ralbidv |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ran (,) ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∈ dom vol ↔ ∀ 𝑦 ∈ ran (,) ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∈ dom vol ) ) |
35 |
4
|
fmpttd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) : 𝐵 ⟶ ℝ ) |
36 |
|
ismbf |
⊢ ( ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) : 𝐵 ⟶ ℝ → ( ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) ∈ MblFn ↔ ∀ 𝑦 ∈ ran (,) ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∈ dom vol ) ) |
37 |
35 36
|
syl |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) ∈ MblFn ↔ ∀ 𝑦 ∈ ran (,) ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) “ 𝑦 ) ∈ dom vol ) ) |
38 |
5
|
fmpttd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) : 𝐵 ⟶ ℝ ) |
39 |
|
ismbf |
⊢ ( ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) : 𝐵 ⟶ ℝ → ( ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) ∈ MblFn ↔ ∀ 𝑦 ∈ ran (,) ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∈ dom vol ) ) |
40 |
38 39
|
syl |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) ∈ MblFn ↔ ∀ 𝑦 ∈ ran (,) ( ◡ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) “ 𝑦 ) ∈ dom vol ) ) |
41 |
34 37 40
|
3bitr4d |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) ∈ MblFn ↔ ( 𝑥 ∈ 𝐵 ↦ 𝐷 ) ∈ MblFn ) ) |