Step |
Hyp |
Ref |
Expression |
1 |
|
unblem.2 |
⊢ 𝐹 = ( rec ( ( 𝑥 ∈ V ↦ ∩ ( 𝐴 ∖ suc 𝑥 ) ) , ∩ 𝐴 ) ↾ ω ) |
2 |
|
fveq2 |
⊢ ( 𝑧 = ∅ → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ ∅ ) ) |
3 |
2
|
eleq1d |
⊢ ( 𝑧 = ∅ → ( ( 𝐹 ‘ 𝑧 ) ∈ 𝐴 ↔ ( 𝐹 ‘ ∅ ) ∈ 𝐴 ) ) |
4 |
|
fveq2 |
⊢ ( 𝑧 = 𝑢 → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑢 ) ) |
5 |
4
|
eleq1d |
⊢ ( 𝑧 = 𝑢 → ( ( 𝐹 ‘ 𝑧 ) ∈ 𝐴 ↔ ( 𝐹 ‘ 𝑢 ) ∈ 𝐴 ) ) |
6 |
|
fveq2 |
⊢ ( 𝑧 = suc 𝑢 → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ suc 𝑢 ) ) |
7 |
6
|
eleq1d |
⊢ ( 𝑧 = suc 𝑢 → ( ( 𝐹 ‘ 𝑧 ) ∈ 𝐴 ↔ ( 𝐹 ‘ suc 𝑢 ) ∈ 𝐴 ) ) |
8 |
|
omsson |
⊢ ω ⊆ On |
9 |
|
sstr |
⊢ ( ( 𝐴 ⊆ ω ∧ ω ⊆ On ) → 𝐴 ⊆ On ) |
10 |
8 9
|
mpan2 |
⊢ ( 𝐴 ⊆ ω → 𝐴 ⊆ On ) |
11 |
|
peano1 |
⊢ ∅ ∈ ω |
12 |
|
eleq1 |
⊢ ( 𝑤 = ∅ → ( 𝑤 ∈ 𝑣 ↔ ∅ ∈ 𝑣 ) ) |
13 |
12
|
rexbidv |
⊢ ( 𝑤 = ∅ → ( ∃ 𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 ↔ ∃ 𝑣 ∈ 𝐴 ∅ ∈ 𝑣 ) ) |
14 |
13
|
rspcv |
⊢ ( ∅ ∈ ω → ( ∀ 𝑤 ∈ ω ∃ 𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → ∃ 𝑣 ∈ 𝐴 ∅ ∈ 𝑣 ) ) |
15 |
11 14
|
ax-mp |
⊢ ( ∀ 𝑤 ∈ ω ∃ 𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → ∃ 𝑣 ∈ 𝐴 ∅ ∈ 𝑣 ) |
16 |
|
df-rex |
⊢ ( ∃ 𝑣 ∈ 𝐴 ∅ ∈ 𝑣 ↔ ∃ 𝑣 ( 𝑣 ∈ 𝐴 ∧ ∅ ∈ 𝑣 ) ) |
17 |
15 16
|
sylib |
⊢ ( ∀ 𝑤 ∈ ω ∃ 𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → ∃ 𝑣 ( 𝑣 ∈ 𝐴 ∧ ∅ ∈ 𝑣 ) ) |
18 |
|
exsimpl |
⊢ ( ∃ 𝑣 ( 𝑣 ∈ 𝐴 ∧ ∅ ∈ 𝑣 ) → ∃ 𝑣 𝑣 ∈ 𝐴 ) |
19 |
17 18
|
syl |
⊢ ( ∀ 𝑤 ∈ ω ∃ 𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → ∃ 𝑣 𝑣 ∈ 𝐴 ) |
20 |
|
n0 |
⊢ ( 𝐴 ≠ ∅ ↔ ∃ 𝑣 𝑣 ∈ 𝐴 ) |
21 |
19 20
|
sylibr |
⊢ ( ∀ 𝑤 ∈ ω ∃ 𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 → 𝐴 ≠ ∅ ) |
22 |
|
onint |
⊢ ( ( 𝐴 ⊆ On ∧ 𝐴 ≠ ∅ ) → ∩ 𝐴 ∈ 𝐴 ) |
23 |
10 21 22
|
syl2an |
⊢ ( ( 𝐴 ⊆ ω ∧ ∀ 𝑤 ∈ ω ∃ 𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 ) → ∩ 𝐴 ∈ 𝐴 ) |
24 |
1
|
fveq1i |
⊢ ( 𝐹 ‘ ∅ ) = ( ( rec ( ( 𝑥 ∈ V ↦ ∩ ( 𝐴 ∖ suc 𝑥 ) ) , ∩ 𝐴 ) ↾ ω ) ‘ ∅ ) |
25 |
|
fr0g |
⊢ ( ∩ 𝐴 ∈ 𝐴 → ( ( rec ( ( 𝑥 ∈ V ↦ ∩ ( 𝐴 ∖ suc 𝑥 ) ) , ∩ 𝐴 ) ↾ ω ) ‘ ∅ ) = ∩ 𝐴 ) |
26 |
24 25
|
eqtr2id |
⊢ ( ∩ 𝐴 ∈ 𝐴 → ∩ 𝐴 = ( 𝐹 ‘ ∅ ) ) |
27 |
26
|
eleq1d |
⊢ ( ∩ 𝐴 ∈ 𝐴 → ( ∩ 𝐴 ∈ 𝐴 ↔ ( 𝐹 ‘ ∅ ) ∈ 𝐴 ) ) |
28 |
27
|
ibi |
⊢ ( ∩ 𝐴 ∈ 𝐴 → ( 𝐹 ‘ ∅ ) ∈ 𝐴 ) |
29 |
23 28
|
syl |
⊢ ( ( 𝐴 ⊆ ω ∧ ∀ 𝑤 ∈ ω ∃ 𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 ) → ( 𝐹 ‘ ∅ ) ∈ 𝐴 ) |
30 |
|
unblem1 |
⊢ ( ( ( 𝐴 ⊆ ω ∧ ∀ 𝑤 ∈ ω ∃ 𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝐴 ) → ∩ ( 𝐴 ∖ suc ( 𝐹 ‘ 𝑢 ) ) ∈ 𝐴 ) |
31 |
|
suceq |
⊢ ( 𝑦 = 𝑥 → suc 𝑦 = suc 𝑥 ) |
32 |
31
|
difeq2d |
⊢ ( 𝑦 = 𝑥 → ( 𝐴 ∖ suc 𝑦 ) = ( 𝐴 ∖ suc 𝑥 ) ) |
33 |
32
|
inteqd |
⊢ ( 𝑦 = 𝑥 → ∩ ( 𝐴 ∖ suc 𝑦 ) = ∩ ( 𝐴 ∖ suc 𝑥 ) ) |
34 |
|
suceq |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑢 ) → suc 𝑦 = suc ( 𝐹 ‘ 𝑢 ) ) |
35 |
34
|
difeq2d |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑢 ) → ( 𝐴 ∖ suc 𝑦 ) = ( 𝐴 ∖ suc ( 𝐹 ‘ 𝑢 ) ) ) |
36 |
35
|
inteqd |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑢 ) → ∩ ( 𝐴 ∖ suc 𝑦 ) = ∩ ( 𝐴 ∖ suc ( 𝐹 ‘ 𝑢 ) ) ) |
37 |
1 33 36
|
frsucmpt2 |
⊢ ( ( 𝑢 ∈ ω ∧ ∩ ( 𝐴 ∖ suc ( 𝐹 ‘ 𝑢 ) ) ∈ 𝐴 ) → ( 𝐹 ‘ suc 𝑢 ) = ∩ ( 𝐴 ∖ suc ( 𝐹 ‘ 𝑢 ) ) ) |
38 |
37
|
eqcomd |
⊢ ( ( 𝑢 ∈ ω ∧ ∩ ( 𝐴 ∖ suc ( 𝐹 ‘ 𝑢 ) ) ∈ 𝐴 ) → ∩ ( 𝐴 ∖ suc ( 𝐹 ‘ 𝑢 ) ) = ( 𝐹 ‘ suc 𝑢 ) ) |
39 |
38
|
eleq1d |
⊢ ( ( 𝑢 ∈ ω ∧ ∩ ( 𝐴 ∖ suc ( 𝐹 ‘ 𝑢 ) ) ∈ 𝐴 ) → ( ∩ ( 𝐴 ∖ suc ( 𝐹 ‘ 𝑢 ) ) ∈ 𝐴 ↔ ( 𝐹 ‘ suc 𝑢 ) ∈ 𝐴 ) ) |
40 |
39
|
ex |
⊢ ( 𝑢 ∈ ω → ( ∩ ( 𝐴 ∖ suc ( 𝐹 ‘ 𝑢 ) ) ∈ 𝐴 → ( ∩ ( 𝐴 ∖ suc ( 𝐹 ‘ 𝑢 ) ) ∈ 𝐴 ↔ ( 𝐹 ‘ suc 𝑢 ) ∈ 𝐴 ) ) ) |
41 |
40
|
ibd |
⊢ ( 𝑢 ∈ ω → ( ∩ ( 𝐴 ∖ suc ( 𝐹 ‘ 𝑢 ) ) ∈ 𝐴 → ( 𝐹 ‘ suc 𝑢 ) ∈ 𝐴 ) ) |
42 |
30 41
|
syl5 |
⊢ ( 𝑢 ∈ ω → ( ( ( 𝐴 ⊆ ω ∧ ∀ 𝑤 ∈ ω ∃ 𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 ) ∧ ( 𝐹 ‘ 𝑢 ) ∈ 𝐴 ) → ( 𝐹 ‘ suc 𝑢 ) ∈ 𝐴 ) ) |
43 |
42
|
expd |
⊢ ( 𝑢 ∈ ω → ( ( 𝐴 ⊆ ω ∧ ∀ 𝑤 ∈ ω ∃ 𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 ) → ( ( 𝐹 ‘ 𝑢 ) ∈ 𝐴 → ( 𝐹 ‘ suc 𝑢 ) ∈ 𝐴 ) ) ) |
44 |
3 5 7 29 43
|
finds2 |
⊢ ( 𝑧 ∈ ω → ( ( 𝐴 ⊆ ω ∧ ∀ 𝑤 ∈ ω ∃ 𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝐴 ) ) |
45 |
44
|
com12 |
⊢ ( ( 𝐴 ⊆ ω ∧ ∀ 𝑤 ∈ ω ∃ 𝑣 ∈ 𝐴 𝑤 ∈ 𝑣 ) → ( 𝑧 ∈ ω → ( 𝐹 ‘ 𝑧 ) ∈ 𝐴 ) ) |