Step |
Hyp |
Ref |
Expression |
1 |
|
vex |
⊢ 𝑦 ∈ V |
2 |
1
|
elint2 |
⊢ ( 𝑦 ∈ ∩ { 𝑥 ∣ ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑎 “ 𝐵 ) } ↔ ∀ 𝑧 ∈ { 𝑥 ∣ ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑎 “ 𝐵 ) } 𝑦 ∈ 𝑧 ) |
3 |
|
elequ2 |
⊢ ( 𝑧 = 𝑥 → ( 𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑥 ) ) |
4 |
3
|
ralab2 |
⊢ ( ∀ 𝑧 ∈ { 𝑥 ∣ ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑎 “ 𝐵 ) } 𝑦 ∈ 𝑧 ↔ ∀ 𝑥 ( ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑎 “ 𝐵 ) → 𝑦 ∈ 𝑥 ) ) |
5 |
|
df-rex |
⊢ ( ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑎 “ 𝐵 ) ↔ ∃ 𝑎 ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) ) |
6 |
5
|
imbi1i |
⊢ ( ( ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑎 “ 𝐵 ) → 𝑦 ∈ 𝑥 ) ↔ ( ∃ 𝑎 ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → 𝑦 ∈ 𝑥 ) ) |
7 |
|
19.23v |
⊢ ( ∀ 𝑎 ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → 𝑦 ∈ 𝑥 ) ↔ ( ∃ 𝑎 ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → 𝑦 ∈ 𝑥 ) ) |
8 |
|
simpr |
⊢ ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → 𝑥 = ( 𝑎 “ 𝐵 ) ) |
9 |
8
|
eleq2d |
⊢ ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → ( 𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ( 𝑎 “ 𝐵 ) ) ) |
10 |
9
|
pm5.74i |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → 𝑦 ∈ 𝑥 ) ↔ ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → 𝑦 ∈ ( 𝑎 “ 𝐵 ) ) ) |
11 |
1
|
elima |
⊢ ( 𝑦 ∈ ( 𝑎 “ 𝐵 ) ↔ ∃ 𝑏 ∈ 𝐵 𝑏 𝑎 𝑦 ) |
12 |
|
df-br |
⊢ ( 𝑏 𝑎 𝑦 ↔ 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) |
13 |
12
|
rexbii |
⊢ ( ∃ 𝑏 ∈ 𝐵 𝑏 𝑎 𝑦 ↔ ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) |
14 |
11 13
|
bitri |
⊢ ( 𝑦 ∈ ( 𝑎 “ 𝐵 ) ↔ ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) |
15 |
14
|
imbi2i |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → 𝑦 ∈ ( 𝑎 “ 𝐵 ) ) ↔ ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ) |
16 |
10 15
|
bitri |
⊢ ( ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → 𝑦 ∈ 𝑥 ) ↔ ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ) |
17 |
16
|
albii |
⊢ ( ∀ 𝑎 ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑎 ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ) |
18 |
6 7 17
|
3bitr2i |
⊢ ( ( ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑎 “ 𝐵 ) → 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑎 ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ) |
19 |
18
|
albii |
⊢ ( ∀ 𝑥 ( ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑎 “ 𝐵 ) → 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑥 ∀ 𝑎 ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ) |
20 |
|
19.23v |
⊢ ( ∀ 𝑥 ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ↔ ( ∃ 𝑥 ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ) |
21 |
|
vex |
⊢ 𝑎 ∈ V |
22 |
21
|
imaex |
⊢ ( 𝑎 “ 𝐵 ) ∈ V |
23 |
22
|
isseti |
⊢ ∃ 𝑥 𝑥 = ( 𝑎 “ 𝐵 ) |
24 |
|
19.42v |
⊢ ( ∃ 𝑥 ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) ↔ ( 𝑎 ∈ 𝐴 ∧ ∃ 𝑥 𝑥 = ( 𝑎 “ 𝐵 ) ) ) |
25 |
23 24
|
mpbiran2 |
⊢ ( ∃ 𝑥 ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) ↔ 𝑎 ∈ 𝐴 ) |
26 |
25
|
imbi1i |
⊢ ( ( ∃ 𝑥 ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ↔ ( 𝑎 ∈ 𝐴 → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ) |
27 |
20 26
|
bitri |
⊢ ( ∀ 𝑥 ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ↔ ( 𝑎 ∈ 𝐴 → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ) |
28 |
27
|
albii |
⊢ ( ∀ 𝑎 ∀ 𝑥 ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ↔ ∀ 𝑎 ( 𝑎 ∈ 𝐴 → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ) |
29 |
|
alcom |
⊢ ( ∀ 𝑥 ∀ 𝑎 ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ↔ ∀ 𝑎 ∀ 𝑥 ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ) |
30 |
|
df-ral |
⊢ ( ∀ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ↔ ∀ 𝑎 ( 𝑎 ∈ 𝐴 → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ) |
31 |
28 29 30
|
3bitr4i |
⊢ ( ∀ 𝑥 ∀ 𝑎 ( ( 𝑎 ∈ 𝐴 ∧ 𝑥 = ( 𝑎 “ 𝐵 ) ) → ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) ↔ ∀ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) |
32 |
19 31
|
bitri |
⊢ ( ∀ 𝑥 ( ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑎 “ 𝐵 ) → 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) |
33 |
4 32
|
bitri |
⊢ ( ∀ 𝑧 ∈ { 𝑥 ∣ ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑎 “ 𝐵 ) } 𝑦 ∈ 𝑧 ↔ ∀ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) |
34 |
2 33
|
bitri |
⊢ ( 𝑦 ∈ ∩ { 𝑥 ∣ ∃ 𝑎 ∈ 𝐴 𝑥 = ( 𝑎 “ 𝐵 ) } ↔ ∀ 𝑎 ∈ 𝐴 ∃ 𝑏 ∈ 𝐵 〈 𝑏 , 𝑦 〉 ∈ 𝑎 ) |