Step |
Hyp |
Ref |
Expression |
1 |
|
limcun.1 |
⊢ ( 𝜑 → 𝐴 ⊆ ℂ ) |
2 |
|
limcun.2 |
⊢ ( 𝜑 → 𝐵 ⊆ ℂ ) |
3 |
|
limcun.3 |
⊢ ( 𝜑 → 𝐹 : ( 𝐴 ∪ 𝐵 ) ⟶ ℂ ) |
4 |
|
limcrcl |
⊢ ( 𝑥 ∈ ( 𝐹 limℂ 𝐶 ) → ( 𝐹 : dom 𝐹 ⟶ ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐶 ∈ ℂ ) ) |
5 |
4
|
simp3d |
⊢ ( 𝑥 ∈ ( 𝐹 limℂ 𝐶 ) → 𝐶 ∈ ℂ ) |
6 |
5
|
a1i |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐹 limℂ 𝐶 ) → 𝐶 ∈ ℂ ) ) |
7 |
|
elinel1 |
⊢ ( 𝑥 ∈ ( ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∩ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) → 𝑥 ∈ ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ) |
8 |
|
limcrcl |
⊢ ( 𝑥 ∈ ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) → ( ( 𝐹 ↾ 𝐴 ) : dom ( 𝐹 ↾ 𝐴 ) ⟶ ℂ ∧ dom ( 𝐹 ↾ 𝐴 ) ⊆ ℂ ∧ 𝐶 ∈ ℂ ) ) |
9 |
8
|
simp3d |
⊢ ( 𝑥 ∈ ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) → 𝐶 ∈ ℂ ) |
10 |
7 9
|
syl |
⊢ ( 𝑥 ∈ ( ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∩ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) → 𝐶 ∈ ℂ ) |
11 |
10
|
a1i |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∩ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) → 𝐶 ∈ ℂ ) ) |
12 |
|
prfi |
⊢ { 𝐴 , 𝐵 } ∈ Fin |
13 |
12
|
a1i |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → { 𝐴 , 𝐵 } ∈ Fin ) |
14 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → 𝐴 ⊆ ℂ ) |
15 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → 𝐵 ⊆ ℂ ) |
16 |
|
cnex |
⊢ ℂ ∈ V |
17 |
16
|
ssex |
⊢ ( 𝐴 ⊆ ℂ → 𝐴 ∈ V ) |
18 |
14 17
|
syl |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → 𝐴 ∈ V ) |
19 |
16
|
ssex |
⊢ ( 𝐵 ⊆ ℂ → 𝐵 ∈ V ) |
20 |
15 19
|
syl |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → 𝐵 ∈ V ) |
21 |
|
sseq1 |
⊢ ( 𝑦 = 𝐴 → ( 𝑦 ⊆ ℂ ↔ 𝐴 ⊆ ℂ ) ) |
22 |
|
sseq1 |
⊢ ( 𝑦 = 𝐵 → ( 𝑦 ⊆ ℂ ↔ 𝐵 ⊆ ℂ ) ) |
23 |
21 22
|
ralprg |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ∀ 𝑦 ∈ { 𝐴 , 𝐵 } 𝑦 ⊆ ℂ ↔ ( 𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ ) ) ) |
24 |
18 20 23
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → ( ∀ 𝑦 ∈ { 𝐴 , 𝐵 } 𝑦 ⊆ ℂ ↔ ( 𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ ) ) ) |
25 |
14 15 24
|
mpbir2and |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → ∀ 𝑦 ∈ { 𝐴 , 𝐵 } 𝑦 ⊆ ℂ ) |
26 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → 𝐹 : ( 𝐴 ∪ 𝐵 ) ⟶ ℂ ) |
27 |
|
uniiun |
⊢ ∪ { 𝐴 , 𝐵 } = ∪ 𝑦 ∈ { 𝐴 , 𝐵 } 𝑦 |
28 |
|
uniprg |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ∪ { 𝐴 , 𝐵 } = ( 𝐴 ∪ 𝐵 ) ) |
29 |
18 20 28
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → ∪ { 𝐴 , 𝐵 } = ( 𝐴 ∪ 𝐵 ) ) |
30 |
27 29
|
syl5eqr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → ∪ 𝑦 ∈ { 𝐴 , 𝐵 } 𝑦 = ( 𝐴 ∪ 𝐵 ) ) |
31 |
30
|
feq2d |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → ( 𝐹 : ∪ 𝑦 ∈ { 𝐴 , 𝐵 } 𝑦 ⟶ ℂ ↔ 𝐹 : ( 𝐴 ∪ 𝐵 ) ⟶ ℂ ) ) |
32 |
26 31
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → 𝐹 : ∪ 𝑦 ∈ { 𝐴 , 𝐵 } 𝑦 ⟶ ℂ ) |
33 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → 𝐶 ∈ ℂ ) |
34 |
13 25 32 33
|
limciun |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → ( 𝐹 limℂ 𝐶 ) = ( ℂ ∩ ∩ 𝑦 ∈ { 𝐴 , 𝐵 } ( ( 𝐹 ↾ 𝑦 ) limℂ 𝐶 ) ) ) |
35 |
34
|
eleq2d |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → ( 𝑥 ∈ ( 𝐹 limℂ 𝐶 ) ↔ 𝑥 ∈ ( ℂ ∩ ∩ 𝑦 ∈ { 𝐴 , 𝐵 } ( ( 𝐹 ↾ 𝑦 ) limℂ 𝐶 ) ) ) ) |
36 |
|
reseq2 |
⊢ ( 𝑦 = 𝐴 → ( 𝐹 ↾ 𝑦 ) = ( 𝐹 ↾ 𝐴 ) ) |
37 |
36
|
oveq1d |
⊢ ( 𝑦 = 𝐴 → ( ( 𝐹 ↾ 𝑦 ) limℂ 𝐶 ) = ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ) |
38 |
37
|
eleq2d |
⊢ ( 𝑦 = 𝐴 → ( 𝑥 ∈ ( ( 𝐹 ↾ 𝑦 ) limℂ 𝐶 ) ↔ 𝑥 ∈ ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ) ) |
39 |
|
reseq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝐹 ↾ 𝑦 ) = ( 𝐹 ↾ 𝐵 ) ) |
40 |
39
|
oveq1d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝐹 ↾ 𝑦 ) limℂ 𝐶 ) = ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) |
41 |
40
|
eleq2d |
⊢ ( 𝑦 = 𝐵 → ( 𝑥 ∈ ( ( 𝐹 ↾ 𝑦 ) limℂ 𝐶 ) ↔ 𝑥 ∈ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) ) |
42 |
38 41
|
ralprg |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ∀ 𝑦 ∈ { 𝐴 , 𝐵 } 𝑥 ∈ ( ( 𝐹 ↾ 𝑦 ) limℂ 𝐶 ) ↔ ( 𝑥 ∈ ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∧ 𝑥 ∈ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) ) ) |
43 |
18 20 42
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → ( ∀ 𝑦 ∈ { 𝐴 , 𝐵 } 𝑥 ∈ ( ( 𝐹 ↾ 𝑦 ) limℂ 𝐶 ) ↔ ( 𝑥 ∈ ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∧ 𝑥 ∈ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) ) ) |
44 |
43
|
anbi2d |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → ( ( 𝑥 ∈ ℂ ∧ ∀ 𝑦 ∈ { 𝐴 , 𝐵 } 𝑥 ∈ ( ( 𝐹 ↾ 𝑦 ) limℂ 𝐶 ) ) ↔ ( 𝑥 ∈ ℂ ∧ ( 𝑥 ∈ ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∧ 𝑥 ∈ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) ) ) ) |
45 |
|
limccl |
⊢ ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ⊆ ℂ |
46 |
45
|
sseli |
⊢ ( 𝑥 ∈ ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) → 𝑥 ∈ ℂ ) |
47 |
46
|
adantr |
⊢ ( ( 𝑥 ∈ ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∧ 𝑥 ∈ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) → 𝑥 ∈ ℂ ) |
48 |
47
|
pm4.71ri |
⊢ ( ( 𝑥 ∈ ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∧ 𝑥 ∈ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) ↔ ( 𝑥 ∈ ℂ ∧ ( 𝑥 ∈ ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∧ 𝑥 ∈ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) ) ) |
49 |
44 48
|
bitr4di |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → ( ( 𝑥 ∈ ℂ ∧ ∀ 𝑦 ∈ { 𝐴 , 𝐵 } 𝑥 ∈ ( ( 𝐹 ↾ 𝑦 ) limℂ 𝐶 ) ) ↔ ( 𝑥 ∈ ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∧ 𝑥 ∈ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) ) ) |
50 |
|
elriin |
⊢ ( 𝑥 ∈ ( ℂ ∩ ∩ 𝑦 ∈ { 𝐴 , 𝐵 } ( ( 𝐹 ↾ 𝑦 ) limℂ 𝐶 ) ) ↔ ( 𝑥 ∈ ℂ ∧ ∀ 𝑦 ∈ { 𝐴 , 𝐵 } 𝑥 ∈ ( ( 𝐹 ↾ 𝑦 ) limℂ 𝐶 ) ) ) |
51 |
|
elin |
⊢ ( 𝑥 ∈ ( ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∩ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) ↔ ( 𝑥 ∈ ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∧ 𝑥 ∈ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) ) |
52 |
49 50 51
|
3bitr4g |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → ( 𝑥 ∈ ( ℂ ∩ ∩ 𝑦 ∈ { 𝐴 , 𝐵 } ( ( 𝐹 ↾ 𝑦 ) limℂ 𝐶 ) ) ↔ 𝑥 ∈ ( ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∩ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) ) ) |
53 |
35 52
|
bitrd |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℂ ) → ( 𝑥 ∈ ( 𝐹 limℂ 𝐶 ) ↔ 𝑥 ∈ ( ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∩ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) ) ) |
54 |
53
|
ex |
⊢ ( 𝜑 → ( 𝐶 ∈ ℂ → ( 𝑥 ∈ ( 𝐹 limℂ 𝐶 ) ↔ 𝑥 ∈ ( ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∩ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) ) ) ) |
55 |
6 11 54
|
pm5.21ndd |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐹 limℂ 𝐶 ) ↔ 𝑥 ∈ ( ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∩ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) ) ) |
56 |
55
|
eqrdv |
⊢ ( 𝜑 → ( 𝐹 limℂ 𝐶 ) = ( ( ( 𝐹 ↾ 𝐴 ) limℂ 𝐶 ) ∩ ( ( 𝐹 ↾ 𝐵 ) limℂ 𝐶 ) ) ) |