| Step |
Hyp |
Ref |
Expression |
| 1 |
|
cfval |
⊢ ( 𝐴 ∈ On → ( cf ‘ 𝐴 ) = ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) |
| 2 |
|
vex |
⊢ 𝑣 ∈ V |
| 3 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑣 → ( 𝑥 = ( card ‘ 𝑦 ) ↔ 𝑣 = ( card ‘ 𝑦 ) ) ) |
| 4 |
3
|
anbi1d |
⊢ ( 𝑥 = 𝑣 → ( ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ↔ ( 𝑣 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) ) |
| 5 |
4
|
exbidv |
⊢ ( 𝑥 = 𝑣 → ( ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ↔ ∃ 𝑦 ( 𝑣 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) ) |
| 6 |
2 5
|
elab |
⊢ ( 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ↔ ∃ 𝑦 ( 𝑣 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) |
| 7 |
|
fveq2 |
⊢ ( 𝑣 = ( card ‘ 𝑦 ) → ( card ‘ 𝑣 ) = ( card ‘ ( card ‘ 𝑦 ) ) ) |
| 8 |
|
cardidm |
⊢ ( card ‘ ( card ‘ 𝑦 ) ) = ( card ‘ 𝑦 ) |
| 9 |
7 8
|
eqtrdi |
⊢ ( 𝑣 = ( card ‘ 𝑦 ) → ( card ‘ 𝑣 ) = ( card ‘ 𝑦 ) ) |
| 10 |
|
eqeq2 |
⊢ ( 𝑣 = ( card ‘ 𝑦 ) → ( ( card ‘ 𝑣 ) = 𝑣 ↔ ( card ‘ 𝑣 ) = ( card ‘ 𝑦 ) ) ) |
| 11 |
9 10
|
mpbird |
⊢ ( 𝑣 = ( card ‘ 𝑦 ) → ( card ‘ 𝑣 ) = 𝑣 ) |
| 12 |
11
|
adantr |
⊢ ( ( 𝑣 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) → ( card ‘ 𝑣 ) = 𝑣 ) |
| 13 |
12
|
exlimiv |
⊢ ( ∃ 𝑦 ( 𝑣 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) → ( card ‘ 𝑣 ) = 𝑣 ) |
| 14 |
6 13
|
sylbi |
⊢ ( 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } → ( card ‘ 𝑣 ) = 𝑣 ) |
| 15 |
|
cardon |
⊢ ( card ‘ 𝑣 ) ∈ On |
| 16 |
14 15
|
eqeltrrdi |
⊢ ( 𝑣 ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } → 𝑣 ∈ On ) |
| 17 |
16
|
ssriv |
⊢ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ⊆ On |
| 18 |
|
fvex |
⊢ ( cf ‘ 𝐴 ) ∈ V |
| 19 |
1 18
|
eqeltrrdi |
⊢ ( 𝐴 ∈ On → ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ∈ V ) |
| 20 |
|
intex |
⊢ ( { 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 ) ) } ∈ V ) |
| 21 |
19 20
|
sylibr |
⊢ ( 𝐴 ∈ On → { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ≠ ∅ ) |
| 22 |
|
onint |
⊢ ( ( { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ⊆ On ∧ { 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 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) |
| 23 |
17 21 22
|
sylancr |
⊢ ( 𝐴 ∈ On → ∩ { 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 ) ) } ) |
| 24 |
1 23
|
eqeltrd |
⊢ ( 𝐴 ∈ On → ( cf ‘ 𝐴 ) ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ) |
| 25 |
|
fveq2 |
⊢ ( 𝑣 = ( cf ‘ 𝐴 ) → ( card ‘ 𝑣 ) = ( card ‘ ( cf ‘ 𝐴 ) ) ) |
| 26 |
|
id |
⊢ ( 𝑣 = ( cf ‘ 𝐴 ) → 𝑣 = ( cf ‘ 𝐴 ) ) |
| 27 |
25 26
|
eqeq12d |
⊢ ( 𝑣 = ( cf ‘ 𝐴 ) → ( ( card ‘ 𝑣 ) = 𝑣 ↔ ( card ‘ ( cf ‘ 𝐴 ) ) = ( cf ‘ 𝐴 ) ) ) |
| 28 |
27 14
|
vtoclga |
⊢ ( ( cf ‘ 𝐴 ) ∈ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } → ( card ‘ ( cf ‘ 𝐴 ) ) = ( cf ‘ 𝐴 ) ) |
| 29 |
24 28
|
syl |
⊢ ( 𝐴 ∈ On → ( card ‘ ( cf ‘ 𝐴 ) ) = ( cf ‘ 𝐴 ) ) |
| 30 |
|
cff |
⊢ cf : On ⟶ On |
| 31 |
30
|
fdmi |
⊢ dom cf = On |
| 32 |
31
|
eleq2i |
⊢ ( 𝐴 ∈ dom cf ↔ 𝐴 ∈ On ) |
| 33 |
|
ndmfv |
⊢ ( ¬ 𝐴 ∈ dom cf → ( cf ‘ 𝐴 ) = ∅ ) |
| 34 |
32 33
|
sylnbir |
⊢ ( ¬ 𝐴 ∈ On → ( cf ‘ 𝐴 ) = ∅ ) |
| 35 |
|
card0 |
⊢ ( card ‘ ∅ ) = ∅ |
| 36 |
|
fveq2 |
⊢ ( ( cf ‘ 𝐴 ) = ∅ → ( card ‘ ( cf ‘ 𝐴 ) ) = ( card ‘ ∅ ) ) |
| 37 |
|
id |
⊢ ( ( cf ‘ 𝐴 ) = ∅ → ( cf ‘ 𝐴 ) = ∅ ) |
| 38 |
35 36 37
|
3eqtr4a |
⊢ ( ( cf ‘ 𝐴 ) = ∅ → ( card ‘ ( cf ‘ 𝐴 ) ) = ( cf ‘ 𝐴 ) ) |
| 39 |
34 38
|
syl |
⊢ ( ¬ 𝐴 ∈ On → ( card ‘ ( cf ‘ 𝐴 ) ) = ( cf ‘ 𝐴 ) ) |
| 40 |
29 39
|
pm2.61i |
⊢ ( card ‘ ( cf ‘ 𝐴 ) ) = ( cf ‘ 𝐴 ) |