Step |
Hyp |
Ref |
Expression |
1 |
|
cfval |
⊢ ( 𝐴 ∈ On → ( cf ‘ 𝐴 ) = ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) |
2 |
|
dfss3 |
⊢ ( 𝐴 ⊆ ∪ 𝑦 ↔ ∀ 𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 ) |
3 |
|
ssel |
⊢ ( 𝑦 ⊆ 𝐴 → ( 𝑤 ∈ 𝑦 → 𝑤 ∈ 𝐴 ) ) |
4 |
|
onelon |
⊢ ( ( 𝐴 ∈ On ∧ 𝑤 ∈ 𝐴 ) → 𝑤 ∈ On ) |
5 |
4
|
ex |
⊢ ( 𝐴 ∈ On → ( 𝑤 ∈ 𝐴 → 𝑤 ∈ On ) ) |
6 |
3 5
|
sylan9r |
⊢ ( ( 𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴 ) → ( 𝑤 ∈ 𝑦 → 𝑤 ∈ On ) ) |
7 |
|
onelss |
⊢ ( 𝑤 ∈ On → ( 𝑧 ∈ 𝑤 → 𝑧 ⊆ 𝑤 ) ) |
8 |
6 7
|
syl6 |
⊢ ( ( 𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴 ) → ( 𝑤 ∈ 𝑦 → ( 𝑧 ∈ 𝑤 → 𝑧 ⊆ 𝑤 ) ) ) |
9 |
8
|
imdistand |
⊢ ( ( 𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴 ) → ( ( 𝑤 ∈ 𝑦 ∧ 𝑧 ∈ 𝑤 ) → ( 𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤 ) ) ) |
10 |
9
|
ancomsd |
⊢ ( ( 𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴 ) → ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦 ) → ( 𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤 ) ) ) |
11 |
10
|
eximdv |
⊢ ( ( 𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴 ) → ( ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦 ) → ∃ 𝑤 ( 𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤 ) ) ) |
12 |
|
eluni |
⊢ ( 𝑧 ∈ ∪ 𝑦 ↔ ∃ 𝑤 ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦 ) ) |
13 |
|
df-rex |
⊢ ( ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃ 𝑤 ( 𝑤 ∈ 𝑦 ∧ 𝑧 ⊆ 𝑤 ) ) |
14 |
11 12 13
|
3imtr4g |
⊢ ( ( 𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴 ) → ( 𝑧 ∈ ∪ 𝑦 → ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) |
15 |
14
|
ralimdv |
⊢ ( ( 𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴 ) → ( ∀ 𝑧 ∈ 𝐴 𝑧 ∈ ∪ 𝑦 → ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) |
16 |
2 15
|
syl5bi |
⊢ ( ( 𝐴 ∈ On ∧ 𝑦 ⊆ 𝐴 ) → ( 𝐴 ⊆ ∪ 𝑦 → ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) |
17 |
16
|
imdistanda |
⊢ ( 𝐴 ∈ On → ( ( 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦 ) → ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) |
18 |
17
|
anim2d |
⊢ ( 𝐴 ∈ On → ( ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦 ) ) → ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) ) |
19 |
18
|
eximdv |
⊢ ( 𝐴 ∈ On → ( ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦 ) ) → ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) ) |
20 |
19
|
ss2abdv |
⊢ ( 𝐴 ∈ On → { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦 ) ) } ⊆ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) |
21 |
|
intss |
⊢ ( { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦 ) ) } ⊆ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } → ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ⊆ ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦 ) ) } ) |
22 |
20 21
|
syl |
⊢ ( 𝐴 ∈ On → ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ⊆ ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦 ) ) } ) |
23 |
1 22
|
eqsstrd |
⊢ ( 𝐴 ∈ On → ( cf ‘ 𝐴 ) ⊆ ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦 ) ) } ) |
24 |
|
cff |
⊢ cf : On ⟶ On |
25 |
24
|
fdmi |
⊢ dom cf = On |
26 |
25
|
eleq2i |
⊢ ( 𝐴 ∈ dom cf ↔ 𝐴 ∈ On ) |
27 |
|
ndmfv |
⊢ ( ¬ 𝐴 ∈ dom cf → ( cf ‘ 𝐴 ) = ∅ ) |
28 |
26 27
|
sylnbir |
⊢ ( ¬ 𝐴 ∈ On → ( cf ‘ 𝐴 ) = ∅ ) |
29 |
|
0ss |
⊢ ∅ ⊆ ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦 ) ) } |
30 |
28 29
|
eqsstrdi |
⊢ ( ¬ 𝐴 ∈ On → ( cf ‘ 𝐴 ) ⊆ ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦 ) ) } ) |
31 |
23 30
|
pm2.61i |
⊢ ( cf ‘ 𝐴 ) ⊆ ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦 ) ) } |