Step |
Hyp |
Ref |
Expression |
1 |
|
isfi |
⊢ ( 𝑧 ∈ Fin ↔ ∃ 𝑤 ∈ ω 𝑧 ≈ 𝑤 ) |
2 |
|
nnfi |
⊢ ( 𝑤 ∈ ω → 𝑤 ∈ Fin ) |
3 |
|
ensymfib |
⊢ ( 𝑤 ∈ Fin → ( 𝑤 ≈ 𝑧 ↔ 𝑧 ≈ 𝑤 ) ) |
4 |
2 3
|
syl |
⊢ ( 𝑤 ∈ ω → ( 𝑤 ≈ 𝑧 ↔ 𝑧 ≈ 𝑤 ) ) |
5 |
|
breq1 |
⊢ ( 𝑤 = ∅ → ( 𝑤 ≈ 𝑧 ↔ ∅ ≈ 𝑧 ) ) |
6 |
5
|
anbi2d |
⊢ ( 𝑤 = ∅ → ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑤 ≈ 𝑧 ) ↔ ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ ∅ ≈ 𝑧 ) ) ) |
7 |
6
|
imbi1d |
⊢ ( 𝑤 = ∅ → ( ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑤 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ↔ ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ ∅ ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) ) |
8 |
7
|
albidv |
⊢ ( 𝑤 = ∅ → ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑤 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ↔ ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ ∅ ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) ) |
9 |
|
breq1 |
⊢ ( 𝑤 = 𝑡 → ( 𝑤 ≈ 𝑧 ↔ 𝑡 ≈ 𝑧 ) ) |
10 |
9
|
anbi2d |
⊢ ( 𝑤 = 𝑡 → ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑤 ≈ 𝑧 ) ↔ ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) ) ) |
11 |
10
|
imbi1d |
⊢ ( 𝑤 = 𝑡 → ( ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑤 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ↔ ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) ) |
12 |
11
|
albidv |
⊢ ( 𝑤 = 𝑡 → ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑤 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ↔ ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) ) |
13 |
|
breq1 |
⊢ ( 𝑤 = suc 𝑡 → ( 𝑤 ≈ 𝑧 ↔ suc 𝑡 ≈ 𝑧 ) ) |
14 |
13
|
anbi2d |
⊢ ( 𝑤 = suc 𝑡 → ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑤 ≈ 𝑧 ) ↔ ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ suc 𝑡 ≈ 𝑧 ) ) ) |
15 |
14
|
imbi1d |
⊢ ( 𝑤 = suc 𝑡 → ( ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑤 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ↔ ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ suc 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) ) |
16 |
15
|
albidv |
⊢ ( 𝑤 = suc 𝑡 → ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑤 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ↔ ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ suc 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) ) |
17 |
|
en0r |
⊢ ( ∅ ≈ 𝑧 ↔ 𝑧 = ∅ ) |
18 |
17
|
biimpi |
⊢ ( ∅ ≈ 𝑧 → 𝑧 = ∅ ) |
19 |
18
|
anim1i |
⊢ ( ( ∅ ≈ 𝑧 ∧ 𝑧 ≠ ∅ ) → ( 𝑧 = ∅ ∧ 𝑧 ≠ ∅ ) ) |
20 |
19
|
ancoms |
⊢ ( ( 𝑧 ≠ ∅ ∧ ∅ ≈ 𝑧 ) → ( 𝑧 = ∅ ∧ 𝑧 ≠ ∅ ) ) |
21 |
20
|
adantll |
⊢ ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ ∅ ≈ 𝑧 ) → ( 𝑧 = ∅ ∧ 𝑧 ≠ ∅ ) ) |
22 |
|
df-ne |
⊢ ( 𝑧 ≠ ∅ ↔ ¬ 𝑧 = ∅ ) |
23 |
|
pm3.24 |
⊢ ¬ ( 𝑧 = ∅ ∧ ¬ 𝑧 = ∅ ) |
24 |
23
|
pm2.21i |
⊢ ( ( 𝑧 = ∅ ∧ ¬ 𝑧 = ∅ ) → ∩ 𝑧 ∈ 𝐴 ) |
25 |
22 24
|
sylan2b |
⊢ ( ( 𝑧 = ∅ ∧ 𝑧 ≠ ∅ ) → ∩ 𝑧 ∈ 𝐴 ) |
26 |
21 25
|
syl |
⊢ ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ ∅ ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) |
27 |
26
|
ax-gen |
⊢ ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ ∅ ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) |
28 |
27
|
a1i |
⊢ ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ ∅ ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) |
29 |
|
nfv |
⊢ Ⅎ 𝑧 ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 |
30 |
|
nfa1 |
⊢ Ⅎ 𝑧 ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) |
31 |
|
bren |
⊢ ( suc 𝑡 ≈ 𝑧 ↔ ∃ 𝑓 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 ) |
32 |
|
ssel |
⊢ ( 𝑧 ⊆ 𝐴 → ( ( 𝑓 ‘ 𝑡 ) ∈ 𝑧 → ( 𝑓 ‘ 𝑡 ) ∈ 𝐴 ) ) |
33 |
|
f1of |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → 𝑓 : suc 𝑡 ⟶ 𝑧 ) |
34 |
|
vex |
⊢ 𝑡 ∈ V |
35 |
34
|
sucid |
⊢ 𝑡 ∈ suc 𝑡 |
36 |
|
ffvelcdm |
⊢ ( ( 𝑓 : suc 𝑡 ⟶ 𝑧 ∧ 𝑡 ∈ suc 𝑡 ) → ( 𝑓 ‘ 𝑡 ) ∈ 𝑧 ) |
37 |
33 35 36
|
sylancl |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ( 𝑓 ‘ 𝑡 ) ∈ 𝑧 ) |
38 |
32 37
|
impel |
⊢ ( ( 𝑧 ⊆ 𝐴 ∧ 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 ) → ( 𝑓 ‘ 𝑡 ) ∈ 𝐴 ) |
39 |
38
|
adantr |
⊢ ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 ) ∧ ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ∧ ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 ) ) → ( 𝑓 ‘ 𝑡 ) ∈ 𝐴 ) |
40 |
|
df-ne |
⊢ ( ( 𝑓 “ 𝑡 ) ≠ ∅ ↔ ¬ ( 𝑓 “ 𝑡 ) = ∅ ) |
41 |
|
imassrn |
⊢ ( 𝑓 “ 𝑡 ) ⊆ ran 𝑓 |
42 |
|
dff1o2 |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 ↔ ( 𝑓 Fn suc 𝑡 ∧ Fun ◡ 𝑓 ∧ ran 𝑓 = 𝑧 ) ) |
43 |
42
|
simp3bi |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ran 𝑓 = 𝑧 ) |
44 |
41 43
|
sseqtrid |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ( 𝑓 “ 𝑡 ) ⊆ 𝑧 ) |
45 |
|
sstr2 |
⊢ ( ( 𝑓 “ 𝑡 ) ⊆ 𝑧 → ( 𝑧 ⊆ 𝐴 → ( 𝑓 “ 𝑡 ) ⊆ 𝐴 ) ) |
46 |
44 45
|
syl |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ( 𝑧 ⊆ 𝐴 → ( 𝑓 “ 𝑡 ) ⊆ 𝐴 ) ) |
47 |
46
|
anim1d |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ( ( 𝑧 ⊆ 𝐴 ∧ ( 𝑓 “ 𝑡 ) ≠ ∅ ) → ( ( 𝑓 “ 𝑡 ) ⊆ 𝐴 ∧ ( 𝑓 “ 𝑡 ) ≠ ∅ ) ) ) |
48 |
|
f1of1 |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → 𝑓 : suc 𝑡 –1-1→ 𝑧 ) |
49 |
|
sssucid |
⊢ 𝑡 ⊆ suc 𝑡 |
50 |
|
vex |
⊢ 𝑓 ∈ V |
51 |
|
f1imaen3g |
⊢ ( ( 𝑓 : suc 𝑡 –1-1→ 𝑧 ∧ 𝑡 ⊆ suc 𝑡 ∧ 𝑓 ∈ V ) → 𝑡 ≈ ( 𝑓 “ 𝑡 ) ) |
52 |
49 50 51
|
mp3an23 |
⊢ ( 𝑓 : suc 𝑡 –1-1→ 𝑧 → 𝑡 ≈ ( 𝑓 “ 𝑡 ) ) |
53 |
48 52
|
syl |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → 𝑡 ≈ ( 𝑓 “ 𝑡 ) ) |
54 |
47 53
|
jctird |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ( ( 𝑧 ⊆ 𝐴 ∧ ( 𝑓 “ 𝑡 ) ≠ ∅ ) → ( ( ( 𝑓 “ 𝑡 ) ⊆ 𝐴 ∧ ( 𝑓 “ 𝑡 ) ≠ ∅ ) ∧ 𝑡 ≈ ( 𝑓 “ 𝑡 ) ) ) ) |
55 |
50
|
imaex |
⊢ ( 𝑓 “ 𝑡 ) ∈ V |
56 |
|
sseq1 |
⊢ ( 𝑧 = ( 𝑓 “ 𝑡 ) → ( 𝑧 ⊆ 𝐴 ↔ ( 𝑓 “ 𝑡 ) ⊆ 𝐴 ) ) |
57 |
|
neeq1 |
⊢ ( 𝑧 = ( 𝑓 “ 𝑡 ) → ( 𝑧 ≠ ∅ ↔ ( 𝑓 “ 𝑡 ) ≠ ∅ ) ) |
58 |
56 57
|
anbi12d |
⊢ ( 𝑧 = ( 𝑓 “ 𝑡 ) → ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ↔ ( ( 𝑓 “ 𝑡 ) ⊆ 𝐴 ∧ ( 𝑓 “ 𝑡 ) ≠ ∅ ) ) ) |
59 |
|
breq2 |
⊢ ( 𝑧 = ( 𝑓 “ 𝑡 ) → ( 𝑡 ≈ 𝑧 ↔ 𝑡 ≈ ( 𝑓 “ 𝑡 ) ) ) |
60 |
58 59
|
anbi12d |
⊢ ( 𝑧 = ( 𝑓 “ 𝑡 ) → ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) ↔ ( ( ( 𝑓 “ 𝑡 ) ⊆ 𝐴 ∧ ( 𝑓 “ 𝑡 ) ≠ ∅ ) ∧ 𝑡 ≈ ( 𝑓 “ 𝑡 ) ) ) ) |
61 |
|
inteq |
⊢ ( 𝑧 = ( 𝑓 “ 𝑡 ) → ∩ 𝑧 = ∩ ( 𝑓 “ 𝑡 ) ) |
62 |
61
|
eleq1d |
⊢ ( 𝑧 = ( 𝑓 “ 𝑡 ) → ( ∩ 𝑧 ∈ 𝐴 ↔ ∩ ( 𝑓 “ 𝑡 ) ∈ 𝐴 ) ) |
63 |
60 62
|
imbi12d |
⊢ ( 𝑧 = ( 𝑓 “ 𝑡 ) → ( ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ↔ ( ( ( ( 𝑓 “ 𝑡 ) ⊆ 𝐴 ∧ ( 𝑓 “ 𝑡 ) ≠ ∅ ) ∧ 𝑡 ≈ ( 𝑓 “ 𝑡 ) ) → ∩ ( 𝑓 “ 𝑡 ) ∈ 𝐴 ) ) ) |
64 |
55 63
|
spcv |
⊢ ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) → ( ( ( ( 𝑓 “ 𝑡 ) ⊆ 𝐴 ∧ ( 𝑓 “ 𝑡 ) ≠ ∅ ) ∧ 𝑡 ≈ ( 𝑓 “ 𝑡 ) ) → ∩ ( 𝑓 “ 𝑡 ) ∈ 𝐴 ) ) |
65 |
54 64
|
sylan9 |
⊢ ( ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 ∧ ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) → ( ( 𝑧 ⊆ 𝐴 ∧ ( 𝑓 “ 𝑡 ) ≠ ∅ ) → ∩ ( 𝑓 “ 𝑡 ) ∈ 𝐴 ) ) |
66 |
|
ineq1 |
⊢ ( 𝑣 = ∩ ( 𝑓 “ 𝑡 ) → ( 𝑣 ∩ 𝑢 ) = ( ∩ ( 𝑓 “ 𝑡 ) ∩ 𝑢 ) ) |
67 |
66
|
eleq1d |
⊢ ( 𝑣 = ∩ ( 𝑓 “ 𝑡 ) → ( ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 ↔ ( ∩ ( 𝑓 “ 𝑡 ) ∩ 𝑢 ) ∈ 𝐴 ) ) |
68 |
|
ineq2 |
⊢ ( 𝑢 = ( 𝑓 ‘ 𝑡 ) → ( ∩ ( 𝑓 “ 𝑡 ) ∩ 𝑢 ) = ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ) |
69 |
68
|
eleq1d |
⊢ ( 𝑢 = ( 𝑓 ‘ 𝑡 ) → ( ( ∩ ( 𝑓 “ 𝑡 ) ∩ 𝑢 ) ∈ 𝐴 ↔ ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ∈ 𝐴 ) ) |
70 |
67 69
|
rspc2v |
⊢ ( ( ∩ ( 𝑓 “ 𝑡 ) ∈ 𝐴 ∧ ( 𝑓 ‘ 𝑡 ) ∈ 𝐴 ) → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ∈ 𝐴 ) ) |
71 |
70
|
ex |
⊢ ( ∩ ( 𝑓 “ 𝑡 ) ∈ 𝐴 → ( ( 𝑓 ‘ 𝑡 ) ∈ 𝐴 → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ∈ 𝐴 ) ) ) |
72 |
65 71
|
syl6 |
⊢ ( ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 ∧ ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) → ( ( 𝑧 ⊆ 𝐴 ∧ ( 𝑓 “ 𝑡 ) ≠ ∅ ) → ( ( 𝑓 ‘ 𝑡 ) ∈ 𝐴 → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ∈ 𝐴 ) ) ) ) |
73 |
72
|
com4r |
⊢ ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ( ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 ∧ ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) → ( ( 𝑧 ⊆ 𝐴 ∧ ( 𝑓 “ 𝑡 ) ≠ ∅ ) → ( ( 𝑓 ‘ 𝑡 ) ∈ 𝐴 → ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ∈ 𝐴 ) ) ) ) |
74 |
73
|
exp5c |
⊢ ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) → ( 𝑧 ⊆ 𝐴 → ( ( 𝑓 “ 𝑡 ) ≠ ∅ → ( ( 𝑓 ‘ 𝑡 ) ∈ 𝐴 → ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ∈ 𝐴 ) ) ) ) ) ) |
75 |
74
|
com14 |
⊢ ( 𝑧 ⊆ 𝐴 → ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ( ( 𝑓 “ 𝑡 ) ≠ ∅ → ( ( 𝑓 ‘ 𝑡 ) ∈ 𝐴 → ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ∈ 𝐴 ) ) ) ) ) ) |
76 |
75
|
imp43 |
⊢ ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 ) ∧ ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ∧ ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 ) ) → ( ( 𝑓 “ 𝑡 ) ≠ ∅ → ( ( 𝑓 ‘ 𝑡 ) ∈ 𝐴 → ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ∈ 𝐴 ) ) ) |
77 |
40 76
|
biimtrrid |
⊢ ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 ) ∧ ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ∧ ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 ) ) → ( ¬ ( 𝑓 “ 𝑡 ) = ∅ → ( ( 𝑓 ‘ 𝑡 ) ∈ 𝐴 → ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ∈ 𝐴 ) ) ) |
78 |
|
inteq |
⊢ ( ( 𝑓 “ 𝑡 ) = ∅ → ∩ ( 𝑓 “ 𝑡 ) = ∩ ∅ ) |
79 |
|
int0 |
⊢ ∩ ∅ = V |
80 |
78 79
|
eqtrdi |
⊢ ( ( 𝑓 “ 𝑡 ) = ∅ → ∩ ( 𝑓 “ 𝑡 ) = V ) |
81 |
80
|
ineq1d |
⊢ ( ( 𝑓 “ 𝑡 ) = ∅ → ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) = ( V ∩ ( 𝑓 ‘ 𝑡 ) ) ) |
82 |
|
ssv |
⊢ ( 𝑓 ‘ 𝑡 ) ⊆ V |
83 |
|
sseqin2 |
⊢ ( ( 𝑓 ‘ 𝑡 ) ⊆ V ↔ ( V ∩ ( 𝑓 ‘ 𝑡 ) ) = ( 𝑓 ‘ 𝑡 ) ) |
84 |
82 83
|
mpbi |
⊢ ( V ∩ ( 𝑓 ‘ 𝑡 ) ) = ( 𝑓 ‘ 𝑡 ) |
85 |
81 84
|
eqtrdi |
⊢ ( ( 𝑓 “ 𝑡 ) = ∅ → ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) = ( 𝑓 ‘ 𝑡 ) ) |
86 |
85
|
eleq1d |
⊢ ( ( 𝑓 “ 𝑡 ) = ∅ → ( ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ∈ 𝐴 ↔ ( 𝑓 ‘ 𝑡 ) ∈ 𝐴 ) ) |
87 |
86
|
biimprd |
⊢ ( ( 𝑓 “ 𝑡 ) = ∅ → ( ( 𝑓 ‘ 𝑡 ) ∈ 𝐴 → ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ∈ 𝐴 ) ) |
88 |
77 87
|
pm2.61d2 |
⊢ ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 ) ∧ ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ∧ ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 ) ) → ( ( 𝑓 ‘ 𝑡 ) ∈ 𝐴 → ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ∈ 𝐴 ) ) |
89 |
39 88
|
mpd |
⊢ ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 ) ∧ ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ∧ ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 ) ) → ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ∈ 𝐴 ) |
90 |
|
fvex |
⊢ ( 𝑓 ‘ 𝑡 ) ∈ V |
91 |
90
|
intunsn |
⊢ ∩ ( ( 𝑓 “ 𝑡 ) ∪ { ( 𝑓 ‘ 𝑡 ) } ) = ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) |
92 |
|
f1ofn |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → 𝑓 Fn suc 𝑡 ) |
93 |
|
fnsnfv |
⊢ ( ( 𝑓 Fn suc 𝑡 ∧ 𝑡 ∈ suc 𝑡 ) → { ( 𝑓 ‘ 𝑡 ) } = ( 𝑓 “ { 𝑡 } ) ) |
94 |
92 35 93
|
sylancl |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → { ( 𝑓 ‘ 𝑡 ) } = ( 𝑓 “ { 𝑡 } ) ) |
95 |
94
|
uneq2d |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ( ( 𝑓 “ 𝑡 ) ∪ { ( 𝑓 ‘ 𝑡 ) } ) = ( ( 𝑓 “ 𝑡 ) ∪ ( 𝑓 “ { 𝑡 } ) ) ) |
96 |
|
df-suc |
⊢ suc 𝑡 = ( 𝑡 ∪ { 𝑡 } ) |
97 |
96
|
imaeq2i |
⊢ ( 𝑓 “ suc 𝑡 ) = ( 𝑓 “ ( 𝑡 ∪ { 𝑡 } ) ) |
98 |
|
imaundi |
⊢ ( 𝑓 “ ( 𝑡 ∪ { 𝑡 } ) ) = ( ( 𝑓 “ 𝑡 ) ∪ ( 𝑓 “ { 𝑡 } ) ) |
99 |
97 98
|
eqtr2i |
⊢ ( ( 𝑓 “ 𝑡 ) ∪ ( 𝑓 “ { 𝑡 } ) ) = ( 𝑓 “ suc 𝑡 ) |
100 |
95 99
|
eqtrdi |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ( ( 𝑓 “ 𝑡 ) ∪ { ( 𝑓 ‘ 𝑡 ) } ) = ( 𝑓 “ suc 𝑡 ) ) |
101 |
|
f1ofo |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → 𝑓 : suc 𝑡 –onto→ 𝑧 ) |
102 |
|
foima |
⊢ ( 𝑓 : suc 𝑡 –onto→ 𝑧 → ( 𝑓 “ suc 𝑡 ) = 𝑧 ) |
103 |
101 102
|
syl |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ( 𝑓 “ suc 𝑡 ) = 𝑧 ) |
104 |
100 103
|
eqtrd |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ( ( 𝑓 “ 𝑡 ) ∪ { ( 𝑓 ‘ 𝑡 ) } ) = 𝑧 ) |
105 |
104
|
inteqd |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ∩ ( ( 𝑓 “ 𝑡 ) ∪ { ( 𝑓 ‘ 𝑡 ) } ) = ∩ 𝑧 ) |
106 |
91 105
|
eqtr3id |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) = ∩ 𝑧 ) |
107 |
106
|
eleq1d |
⊢ ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ( ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ∈ 𝐴 ↔ ∩ 𝑧 ∈ 𝐴 ) ) |
108 |
107
|
ad2antlr |
⊢ ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 ) ∧ ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ∧ ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 ) ) → ( ( ∩ ( 𝑓 “ 𝑡 ) ∩ ( 𝑓 ‘ 𝑡 ) ) ∈ 𝐴 ↔ ∩ 𝑧 ∈ 𝐴 ) ) |
109 |
89 108
|
mpbid |
⊢ ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 ) ∧ ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ∧ ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 ) ) → ∩ 𝑧 ∈ 𝐴 ) |
110 |
109
|
exp43 |
⊢ ( 𝑧 ⊆ 𝐴 → ( 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ∩ 𝑧 ∈ 𝐴 ) ) ) ) |
111 |
110
|
exlimdv |
⊢ ( 𝑧 ⊆ 𝐴 → ( ∃ 𝑓 𝑓 : suc 𝑡 –1-1-onto→ 𝑧 → ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ∩ 𝑧 ∈ 𝐴 ) ) ) ) |
112 |
31 111
|
biimtrid |
⊢ ( 𝑧 ⊆ 𝐴 → ( suc 𝑡 ≈ 𝑧 → ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ∩ 𝑧 ∈ 𝐴 ) ) ) ) |
113 |
112
|
imp |
⊢ ( ( 𝑧 ⊆ 𝐴 ∧ suc 𝑡 ≈ 𝑧 ) → ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ∩ 𝑧 ∈ 𝐴 ) ) ) |
114 |
113
|
adantlr |
⊢ ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ suc 𝑡 ≈ 𝑧 ) → ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ∩ 𝑧 ∈ 𝐴 ) ) ) |
115 |
114
|
com13 |
⊢ ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) → ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ suc 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) ) |
116 |
29 30 115
|
alrimd |
⊢ ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) → ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ suc 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) ) |
117 |
116
|
a1i |
⊢ ( 𝑡 ∈ ω → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) → ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ suc 𝑡 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) ) ) |
118 |
8 12 16 28 117
|
finds2 |
⊢ ( 𝑤 ∈ ω → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑤 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) ) |
119 |
|
sp |
⊢ ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑤 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) → ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑤 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) |
120 |
118 119
|
syl6 |
⊢ ( 𝑤 ∈ ω → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑤 ≈ 𝑧 ) → ∩ 𝑧 ∈ 𝐴 ) ) ) |
121 |
120
|
exp4a |
⊢ ( 𝑤 ∈ ω → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) → ( 𝑤 ≈ 𝑧 → ∩ 𝑧 ∈ 𝐴 ) ) ) ) |
122 |
121
|
com24 |
⊢ ( 𝑤 ∈ ω → ( 𝑤 ≈ 𝑧 → ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ∩ 𝑧 ∈ 𝐴 ) ) ) ) |
123 |
4 122
|
sylbird |
⊢ ( 𝑤 ∈ ω → ( 𝑧 ≈ 𝑤 → ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ∩ 𝑧 ∈ 𝐴 ) ) ) ) |
124 |
123
|
rexlimiv |
⊢ ( ∃ 𝑤 ∈ ω 𝑧 ≈ 𝑤 → ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ∩ 𝑧 ∈ 𝐴 ) ) ) |
125 |
1 124
|
sylbi |
⊢ ( 𝑧 ∈ Fin → ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) → ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ∩ 𝑧 ∈ 𝐴 ) ) ) |
126 |
125
|
com13 |
⊢ ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) → ( 𝑧 ∈ Fin → ∩ 𝑧 ∈ 𝐴 ) ) ) |
127 |
126
|
impd |
⊢ ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑧 ∈ Fin ) → ∩ 𝑧 ∈ 𝐴 ) ) |
128 |
127
|
alrimiv |
⊢ ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 → ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑧 ∈ Fin ) → ∩ 𝑧 ∈ 𝐴 ) ) |
129 |
|
zfpair2 |
⊢ { 𝑣 , 𝑢 } ∈ V |
130 |
|
sseq1 |
⊢ ( 𝑧 = { 𝑣 , 𝑢 } → ( 𝑧 ⊆ 𝐴 ↔ { 𝑣 , 𝑢 } ⊆ 𝐴 ) ) |
131 |
|
neeq1 |
⊢ ( 𝑧 = { 𝑣 , 𝑢 } → ( 𝑧 ≠ ∅ ↔ { 𝑣 , 𝑢 } ≠ ∅ ) ) |
132 |
130 131
|
anbi12d |
⊢ ( 𝑧 = { 𝑣 , 𝑢 } → ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ↔ ( { 𝑣 , 𝑢 } ⊆ 𝐴 ∧ { 𝑣 , 𝑢 } ≠ ∅ ) ) ) |
133 |
|
eleq1 |
⊢ ( 𝑧 = { 𝑣 , 𝑢 } → ( 𝑧 ∈ Fin ↔ { 𝑣 , 𝑢 } ∈ Fin ) ) |
134 |
132 133
|
anbi12d |
⊢ ( 𝑧 = { 𝑣 , 𝑢 } → ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑧 ∈ Fin ) ↔ ( ( { 𝑣 , 𝑢 } ⊆ 𝐴 ∧ { 𝑣 , 𝑢 } ≠ ∅ ) ∧ { 𝑣 , 𝑢 } ∈ Fin ) ) ) |
135 |
|
inteq |
⊢ ( 𝑧 = { 𝑣 , 𝑢 } → ∩ 𝑧 = ∩ { 𝑣 , 𝑢 } ) |
136 |
135
|
eleq1d |
⊢ ( 𝑧 = { 𝑣 , 𝑢 } → ( ∩ 𝑧 ∈ 𝐴 ↔ ∩ { 𝑣 , 𝑢 } ∈ 𝐴 ) ) |
137 |
134 136
|
imbi12d |
⊢ ( 𝑧 = { 𝑣 , 𝑢 } → ( ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑧 ∈ Fin ) → ∩ 𝑧 ∈ 𝐴 ) ↔ ( ( ( { 𝑣 , 𝑢 } ⊆ 𝐴 ∧ { 𝑣 , 𝑢 } ≠ ∅ ) ∧ { 𝑣 , 𝑢 } ∈ Fin ) → ∩ { 𝑣 , 𝑢 } ∈ 𝐴 ) ) ) |
138 |
129 137
|
spcv |
⊢ ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑧 ∈ Fin ) → ∩ 𝑧 ∈ 𝐴 ) → ( ( ( { 𝑣 , 𝑢 } ⊆ 𝐴 ∧ { 𝑣 , 𝑢 } ≠ ∅ ) ∧ { 𝑣 , 𝑢 } ∈ Fin ) → ∩ { 𝑣 , 𝑢 } ∈ 𝐴 ) ) |
139 |
|
vex |
⊢ 𝑣 ∈ V |
140 |
|
vex |
⊢ 𝑢 ∈ V |
141 |
139 140
|
prss |
⊢ ( ( 𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴 ) ↔ { 𝑣 , 𝑢 } ⊆ 𝐴 ) |
142 |
139
|
prnz |
⊢ { 𝑣 , 𝑢 } ≠ ∅ |
143 |
142
|
biantru |
⊢ ( { 𝑣 , 𝑢 } ⊆ 𝐴 ↔ ( { 𝑣 , 𝑢 } ⊆ 𝐴 ∧ { 𝑣 , 𝑢 } ≠ ∅ ) ) |
144 |
|
prfi |
⊢ { 𝑣 , 𝑢 } ∈ Fin |
145 |
144
|
biantru |
⊢ ( ( { 𝑣 , 𝑢 } ⊆ 𝐴 ∧ { 𝑣 , 𝑢 } ≠ ∅ ) ↔ ( ( { 𝑣 , 𝑢 } ⊆ 𝐴 ∧ { 𝑣 , 𝑢 } ≠ ∅ ) ∧ { 𝑣 , 𝑢 } ∈ Fin ) ) |
146 |
141 143 145
|
3bitrri |
⊢ ( ( ( { 𝑣 , 𝑢 } ⊆ 𝐴 ∧ { 𝑣 , 𝑢 } ≠ ∅ ) ∧ { 𝑣 , 𝑢 } ∈ Fin ) ↔ ( 𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴 ) ) |
147 |
139 140
|
intpr |
⊢ ∩ { 𝑣 , 𝑢 } = ( 𝑣 ∩ 𝑢 ) |
148 |
147
|
eleq1i |
⊢ ( ∩ { 𝑣 , 𝑢 } ∈ 𝐴 ↔ ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 ) |
149 |
138 146 148
|
3imtr3g |
⊢ ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑧 ∈ Fin ) → ∩ 𝑧 ∈ 𝐴 ) → ( ( 𝑣 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴 ) → ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 ) ) |
150 |
149
|
ralrimivv |
⊢ ( ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑧 ∈ Fin ) → ∩ 𝑧 ∈ 𝐴 ) → ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 ) |
151 |
128 150
|
impbii |
⊢ ( ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 ↔ ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑧 ∈ Fin ) → ∩ 𝑧 ∈ 𝐴 ) ) |
152 |
|
ineq1 |
⊢ ( 𝑥 = 𝑣 → ( 𝑥 ∩ 𝑦 ) = ( 𝑣 ∩ 𝑦 ) ) |
153 |
152
|
eleq1d |
⊢ ( 𝑥 = 𝑣 → ( ( 𝑥 ∩ 𝑦 ) ∈ 𝐴 ↔ ( 𝑣 ∩ 𝑦 ) ∈ 𝐴 ) ) |
154 |
|
ineq2 |
⊢ ( 𝑦 = 𝑢 → ( 𝑣 ∩ 𝑦 ) = ( 𝑣 ∩ 𝑢 ) ) |
155 |
154
|
eleq1d |
⊢ ( 𝑦 = 𝑢 → ( ( 𝑣 ∩ 𝑦 ) ∈ 𝐴 ↔ ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 ) ) |
156 |
153 155
|
cbvral2vw |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∩ 𝑦 ) ∈ 𝐴 ↔ ∀ 𝑣 ∈ 𝐴 ∀ 𝑢 ∈ 𝐴 ( 𝑣 ∩ 𝑢 ) ∈ 𝐴 ) |
157 |
|
df-3an |
⊢ ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ 𝑧 ∈ Fin ) ↔ ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑧 ∈ Fin ) ) |
158 |
157
|
imbi1i |
⊢ ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ 𝑧 ∈ Fin ) → ∩ 𝑧 ∈ 𝐴 ) ↔ ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑧 ∈ Fin ) → ∩ 𝑧 ∈ 𝐴 ) ) |
159 |
158
|
albii |
⊢ ( ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ 𝑧 ∈ Fin ) → ∩ 𝑧 ∈ 𝐴 ) ↔ ∀ 𝑧 ( ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ) ∧ 𝑧 ∈ Fin ) → ∩ 𝑧 ∈ 𝐴 ) ) |
160 |
151 156 159
|
3bitr4i |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 ∩ 𝑦 ) ∈ 𝐴 ↔ ∀ 𝑧 ( ( 𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ 𝑧 ∈ Fin ) → ∩ 𝑧 ∈ 𝐴 ) ) |