Step |
Hyp |
Ref |
Expression |
1 |
|
df-iun |
⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 } |
2 |
|
elisset |
⊢ ( 𝐵 ∈ 𝐶 → ∃ 𝑧 𝑧 = 𝐵 ) |
3 |
|
eleq2 |
⊢ ( 𝑧 = 𝐵 → ( 𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝐵 ) ) |
4 |
3
|
pm5.32ri |
⊢ ( ( 𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵 ) ↔ ( 𝑤 ∈ 𝐵 ∧ 𝑧 = 𝐵 ) ) |
5 |
4
|
simplbi2 |
⊢ ( 𝑤 ∈ 𝐵 → ( 𝑧 = 𝐵 → ( 𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵 ) ) ) |
6 |
5
|
eximdv |
⊢ ( 𝑤 ∈ 𝐵 → ( ∃ 𝑧 𝑧 = 𝐵 → ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵 ) ) ) |
7 |
2 6
|
syl5com |
⊢ ( 𝐵 ∈ 𝐶 → ( 𝑤 ∈ 𝐵 → ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵 ) ) ) |
8 |
7
|
ralimi |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∀ 𝑥 ∈ 𝐴 ( 𝑤 ∈ 𝐵 → ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵 ) ) ) |
9 |
|
rexim |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑤 ∈ 𝐵 → ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵 ) ) → ( ∃ 𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵 ) ) ) |
10 |
8 9
|
syl |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ( ∃ 𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵 ) ) ) |
11 |
|
rexcom4 |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵 ) ↔ ∃ 𝑧 ∃ 𝑥 ∈ 𝐴 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵 ) ) |
12 |
|
r19.42v |
⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵 ) ↔ ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) ) |
13 |
12
|
exbii |
⊢ ( ∃ 𝑧 ∃ 𝑥 ∈ 𝐴 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵 ) ↔ ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) ) |
14 |
11 13
|
bitri |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵 ) ↔ ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) ) |
15 |
10 14
|
imbitrdi |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ( ∃ 𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 → ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) ) ) |
16 |
3
|
biimpac |
⊢ ( ( 𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵 ) → 𝑤 ∈ 𝐵 ) |
17 |
16
|
reximi |
⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵 ) → ∃ 𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ) |
18 |
12 17
|
sylbir |
⊢ ( ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) → ∃ 𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ) |
19 |
18
|
exlimiv |
⊢ ( ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) → ∃ 𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ) |
20 |
15 19
|
impbid1 |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ( ∃ 𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) ) ) |
21 |
|
vex |
⊢ 𝑤 ∈ V |
22 |
|
eleq1w |
⊢ ( 𝑧 = 𝑤 → ( 𝑧 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵 ) ) |
23 |
22
|
rexbidv |
⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ) ) |
24 |
21 23
|
elab |
⊢ ( 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 } ↔ ∃ 𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ) |
25 |
|
eluni |
⊢ ( 𝑤 ∈ ∪ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ↔ ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ 𝑧 ∈ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ) ) |
26 |
|
vex |
⊢ 𝑧 ∈ V |
27 |
|
eqeq1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 = 𝐵 ↔ 𝑧 = 𝐵 ) ) |
28 |
27
|
rexbidv |
⊢ ( 𝑦 = 𝑧 → ( ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) ) |
29 |
26 28
|
elab |
⊢ ( 𝑧 ∈ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ↔ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) |
30 |
29
|
anbi2i |
⊢ ( ( 𝑤 ∈ 𝑧 ∧ 𝑧 ∈ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ) ↔ ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) ) |
31 |
30
|
exbii |
⊢ ( ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ 𝑧 ∈ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ) ↔ ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) ) |
32 |
25 31
|
bitri |
⊢ ( 𝑤 ∈ ∪ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ↔ ∃ 𝑧 ( 𝑤 ∈ 𝑧 ∧ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) ) |
33 |
20 24 32
|
3bitr4g |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ( 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 } ↔ 𝑤 ∈ ∪ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ) ) |
34 |
33
|
eqrdv |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 } = ∪ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ) |
35 |
1 34
|
eqtrid |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } ) |