Step |
Hyp |
Ref |
Expression |
1 |
|
i1ff |
⊢ ( 𝐹 ∈ dom ∫1 → 𝐹 : ℝ ⟶ ℝ ) |
2 |
|
ffun |
⊢ ( 𝐹 : ℝ ⟶ ℝ → Fun 𝐹 ) |
3 |
|
inpreima |
⊢ ( Fun 𝐹 → ( ◡ 𝐹 “ ( 𝐴 ∩ ran 𝐹 ) ) = ( ( ◡ 𝐹 “ 𝐴 ) ∩ ( ◡ 𝐹 “ ran 𝐹 ) ) ) |
4 |
|
iunid |
⊢ ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) { 𝑦 } = ( 𝐴 ∩ ran 𝐹 ) |
5 |
4
|
imaeq2i |
⊢ ( ◡ 𝐹 “ ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) { 𝑦 } ) = ( ◡ 𝐹 “ ( 𝐴 ∩ ran 𝐹 ) ) |
6 |
|
imaiun |
⊢ ( ◡ 𝐹 “ ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) { 𝑦 } ) = ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } ) |
7 |
5 6
|
eqtr3i |
⊢ ( ◡ 𝐹 “ ( 𝐴 ∩ ran 𝐹 ) ) = ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } ) |
8 |
|
cnvimass |
⊢ ( ◡ 𝐹 “ 𝐴 ) ⊆ dom 𝐹 |
9 |
|
cnvimarndm |
⊢ ( ◡ 𝐹 “ ran 𝐹 ) = dom 𝐹 |
10 |
8 9
|
sseqtrri |
⊢ ( ◡ 𝐹 “ 𝐴 ) ⊆ ( ◡ 𝐹 “ ran 𝐹 ) |
11 |
|
df-ss |
⊢ ( ( ◡ 𝐹 “ 𝐴 ) ⊆ ( ◡ 𝐹 “ ran 𝐹 ) ↔ ( ( ◡ 𝐹 “ 𝐴 ) ∩ ( ◡ 𝐹 “ ran 𝐹 ) ) = ( ◡ 𝐹 “ 𝐴 ) ) |
12 |
10 11
|
mpbi |
⊢ ( ( ◡ 𝐹 “ 𝐴 ) ∩ ( ◡ 𝐹 “ ran 𝐹 ) ) = ( ◡ 𝐹 “ 𝐴 ) |
13 |
3 7 12
|
3eqtr3g |
⊢ ( Fun 𝐹 → ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } ) = ( ◡ 𝐹 “ 𝐴 ) ) |
14 |
1 2 13
|
3syl |
⊢ ( 𝐹 ∈ dom ∫1 → ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } ) = ( ◡ 𝐹 “ 𝐴 ) ) |
15 |
|
i1frn |
⊢ ( 𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin ) |
16 |
|
inss2 |
⊢ ( 𝐴 ∩ ran 𝐹 ) ⊆ ran 𝐹 |
17 |
|
ssfi |
⊢ ( ( ran 𝐹 ∈ Fin ∧ ( 𝐴 ∩ ran 𝐹 ) ⊆ ran 𝐹 ) → ( 𝐴 ∩ ran 𝐹 ) ∈ Fin ) |
18 |
15 16 17
|
sylancl |
⊢ ( 𝐹 ∈ dom ∫1 → ( 𝐴 ∩ ran 𝐹 ) ∈ Fin ) |
19 |
|
i1fmbf |
⊢ ( 𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn ) |
20 |
19
|
adantr |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ) → 𝐹 ∈ MblFn ) |
21 |
1
|
adantr |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ) → 𝐹 : ℝ ⟶ ℝ ) |
22 |
1
|
frnd |
⊢ ( 𝐹 ∈ dom ∫1 → ran 𝐹 ⊆ ℝ ) |
23 |
16 22
|
sstrid |
⊢ ( 𝐹 ∈ dom ∫1 → ( 𝐴 ∩ ran 𝐹 ) ⊆ ℝ ) |
24 |
23
|
sselda |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ) → 𝑦 ∈ ℝ ) |
25 |
|
mbfimasn |
⊢ ( ( 𝐹 ∈ MblFn ∧ 𝐹 : ℝ ⟶ ℝ ∧ 𝑦 ∈ ℝ ) → ( ◡ 𝐹 “ { 𝑦 } ) ∈ dom vol ) |
26 |
20 21 24 25
|
syl3anc |
⊢ ( ( 𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ) → ( ◡ 𝐹 “ { 𝑦 } ) ∈ dom vol ) |
27 |
26
|
ralrimiva |
⊢ ( 𝐹 ∈ dom ∫1 → ∀ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } ) ∈ dom vol ) |
28 |
|
finiunmbl |
⊢ ( ( ( 𝐴 ∩ ran 𝐹 ) ∈ Fin ∧ ∀ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } ) ∈ dom vol ) → ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } ) ∈ dom vol ) |
29 |
18 27 28
|
syl2anc |
⊢ ( 𝐹 ∈ dom ∫1 → ∪ 𝑦 ∈ ( 𝐴 ∩ ran 𝐹 ) ( ◡ 𝐹 “ { 𝑦 } ) ∈ dom vol ) |
30 |
14 29
|
eqeltrrd |
⊢ ( 𝐹 ∈ dom ∫1 → ( ◡ 𝐹 “ 𝐴 ) ∈ dom vol ) |