Step |
Hyp |
Ref |
Expression |
1 |
|
utoptop.1 |
⊢ 𝐽 = ( unifTop ‘ 𝑈 ) |
2 |
|
utopsnneip.1 |
⊢ 𝐾 = { 𝑎 ∈ 𝒫 𝑋 ∣ ∀ 𝑝 ∈ 𝑎 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) } |
3 |
|
utopsnneip.2 |
⊢ 𝑁 = ( 𝑝 ∈ 𝑋 ↦ ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) ) |
4 |
|
utopval |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( unifTop ‘ 𝑈 ) = { 𝑎 ∈ 𝒫 𝑋 ∣ ∀ 𝑝 ∈ 𝑎 ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 } ) |
5 |
1 4
|
eqtrid |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝐽 = { 𝑎 ∈ 𝒫 𝑋 ∣ ∀ 𝑝 ∈ 𝑎 ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 } ) |
6 |
|
simpll |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) → 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ) |
7 |
|
simpr |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) → 𝑎 ∈ 𝒫 𝑋 ) |
8 |
7
|
elpwid |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) → 𝑎 ⊆ 𝑋 ) |
9 |
8
|
sselda |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) → 𝑝 ∈ 𝑋 ) |
10 |
|
simpr |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) → 𝑝 ∈ 𝑋 ) |
11 |
|
mptexg |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) ∈ V ) |
12 |
|
rnexg |
⊢ ( ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) ∈ V → ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) ∈ V ) |
13 |
11 12
|
syl |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) ∈ V ) |
14 |
13
|
adantr |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) → ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) ∈ V ) |
15 |
3
|
fvmpt2 |
⊢ ( ( 𝑝 ∈ 𝑋 ∧ ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) ∈ V ) → ( 𝑁 ‘ 𝑝 ) = ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) ) |
16 |
10 14 15
|
syl2anc |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) → ( 𝑁 ‘ 𝑝 ) = ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) ) |
17 |
16
|
eleq2d |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) → ( 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ↔ 𝑎 ∈ ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) ) ) |
18 |
|
eqid |
⊢ ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) = ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) |
19 |
18
|
elrnmpt |
⊢ ( 𝑎 ∈ V → ( 𝑎 ∈ ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) ↔ ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) ) ) |
20 |
19
|
elv |
⊢ ( 𝑎 ∈ ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) ↔ ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) ) |
21 |
17 20
|
bitrdi |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) → ( 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ↔ ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) ) ) |
22 |
6 9 21
|
syl2anc |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) → ( 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ↔ ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) ) ) |
23 |
|
nfv |
⊢ Ⅎ 𝑣 ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) |
24 |
|
nfre1 |
⊢ Ⅎ 𝑣 ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) |
25 |
23 24
|
nfan |
⊢ Ⅎ 𝑣 ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) ) |
26 |
|
simplr |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑎 = ( 𝑣 “ { 𝑝 } ) ) → 𝑣 ∈ 𝑈 ) |
27 |
|
eqimss2 |
⊢ ( 𝑎 = ( 𝑣 “ { 𝑝 } ) → ( 𝑣 “ { 𝑝 } ) ⊆ 𝑎 ) |
28 |
27
|
adantl |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑎 = ( 𝑣 “ { 𝑝 } ) ) → ( 𝑣 “ { 𝑝 } ) ⊆ 𝑎 ) |
29 |
|
imaeq1 |
⊢ ( 𝑤 = 𝑣 → ( 𝑤 “ { 𝑝 } ) = ( 𝑣 “ { 𝑝 } ) ) |
30 |
29
|
sseq1d |
⊢ ( 𝑤 = 𝑣 → ( ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ↔ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑎 ) ) |
31 |
30
|
rspcev |
⊢ ( ( 𝑣 ∈ 𝑈 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑎 ) → ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) |
32 |
26 28 31
|
syl2anc |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) ) ∧ 𝑣 ∈ 𝑈 ) ∧ 𝑎 = ( 𝑣 “ { 𝑝 } ) ) → ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) |
33 |
|
simpr |
⊢ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) ) → ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) ) |
34 |
25 32 33
|
r19.29af |
⊢ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) ) → ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) |
35 |
6
|
ad2antrr |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ 𝑤 ∈ 𝑈 ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) → 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ) |
36 |
9
|
ad2antrr |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ 𝑤 ∈ 𝑈 ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) → 𝑝 ∈ 𝑋 ) |
37 |
35 36
|
jca |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ 𝑤 ∈ 𝑈 ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) → ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ) |
38 |
|
simpr |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ 𝑤 ∈ 𝑈 ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) → ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) |
39 |
8
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ 𝑤 ∈ 𝑈 ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) → 𝑎 ⊆ 𝑋 ) |
40 |
|
simplr |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ 𝑤 ∈ 𝑈 ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) → 𝑤 ∈ 𝑈 ) |
41 |
|
eqid |
⊢ ( 𝑤 “ { 𝑝 } ) = ( 𝑤 “ { 𝑝 } ) |
42 |
|
imaeq1 |
⊢ ( 𝑢 = 𝑤 → ( 𝑢 “ { 𝑝 } ) = ( 𝑤 “ { 𝑝 } ) ) |
43 |
42
|
rspceeqv |
⊢ ( ( 𝑤 ∈ 𝑈 ∧ ( 𝑤 “ { 𝑝 } ) = ( 𝑤 “ { 𝑝 } ) ) → ∃ 𝑢 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) = ( 𝑢 “ { 𝑝 } ) ) |
44 |
41 43
|
mpan2 |
⊢ ( 𝑤 ∈ 𝑈 → ∃ 𝑢 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) = ( 𝑢 “ { 𝑝 } ) ) |
45 |
44
|
adantl |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) → ∃ 𝑢 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) = ( 𝑢 “ { 𝑝 } ) ) |
46 |
|
vex |
⊢ 𝑤 ∈ V |
47 |
46
|
imaex |
⊢ ( 𝑤 “ { 𝑝 } ) ∈ V |
48 |
3
|
ustuqtoplem |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ ( 𝑤 “ { 𝑝 } ) ∈ V ) → ( ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑝 ) ↔ ∃ 𝑢 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) = ( 𝑢 “ { 𝑝 } ) ) ) |
49 |
47 48
|
mpan2 |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) → ( ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑝 ) ↔ ∃ 𝑢 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) = ( 𝑢 “ { 𝑝 } ) ) ) |
50 |
49
|
adantr |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) → ( ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑝 ) ↔ ∃ 𝑢 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) = ( 𝑢 “ { 𝑝 } ) ) ) |
51 |
45 50
|
mpbird |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) → ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑝 ) ) |
52 |
35 36 40 51
|
syl21anc |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ 𝑤 ∈ 𝑈 ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) → ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑝 ) ) |
53 |
|
sseq1 |
⊢ ( 𝑏 = ( 𝑤 “ { 𝑝 } ) → ( 𝑏 ⊆ 𝑎 ↔ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) ) |
54 |
53
|
3anbi2d |
⊢ ( 𝑏 = ( 𝑤 “ { 𝑝 } ) → ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑏 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋 ) ↔ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋 ) ) ) |
55 |
|
eleq1 |
⊢ ( 𝑏 = ( 𝑤 “ { 𝑝 } ) → ( 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ↔ ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑝 ) ) ) |
56 |
54 55
|
anbi12d |
⊢ ( 𝑏 = ( 𝑤 “ { 𝑝 } ) → ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑏 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ) ↔ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋 ) ∧ ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑝 ) ) ) ) |
57 |
56
|
imbi1d |
⊢ ( 𝑏 = ( 𝑤 “ { 𝑝 } ) → ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑏 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ) → 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ) ↔ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋 ) ∧ ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑝 ) ) → 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ) ) ) |
58 |
3
|
ustuqtop1 |
⊢ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑏 ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ) → 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ) |
59 |
47 57 58
|
vtocl |
⊢ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ∧ 𝑎 ⊆ 𝑋 ) ∧ ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑝 ) ) → 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ) |
60 |
37 38 39 52 59
|
syl31anc |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ 𝑤 ∈ 𝑈 ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) → 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ) |
61 |
37 21
|
syl |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ 𝑤 ∈ 𝑈 ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) → ( 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ↔ ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) ) ) |
62 |
60 61
|
mpbid |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ 𝑤 ∈ 𝑈 ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) → ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) ) |
63 |
62
|
r19.29an |
⊢ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) ∧ ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) → ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) ) |
64 |
34 63
|
impbida |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) → ( ∃ 𝑣 ∈ 𝑈 𝑎 = ( 𝑣 “ { 𝑝 } ) ↔ ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) ) |
65 |
22 64
|
bitrd |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) ∧ 𝑝 ∈ 𝑎 ) → ( 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ↔ ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) ) |
66 |
65
|
ralbidva |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑎 ∈ 𝒫 𝑋 ) → ( ∀ 𝑝 ∈ 𝑎 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ↔ ∀ 𝑝 ∈ 𝑎 ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 ) ) |
67 |
66
|
rabbidva |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → { 𝑎 ∈ 𝒫 𝑋 ∣ ∀ 𝑝 ∈ 𝑎 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) } = { 𝑎 ∈ 𝒫 𝑋 ∣ ∀ 𝑝 ∈ 𝑎 ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ 𝑎 } ) |
68 |
5 67
|
eqtr4d |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝐽 = { 𝑎 ∈ 𝒫 𝑋 ∣ ∀ 𝑝 ∈ 𝑎 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) } ) |
69 |
68 2
|
eqtr4di |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝐽 = 𝐾 ) |
70 |
69
|
fveq2d |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( nei ‘ 𝐽 ) = ( nei ‘ 𝐾 ) ) |
71 |
70
|
fveq1d |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) = ( ( nei ‘ 𝐾 ) ‘ { 𝑃 } ) ) |
72 |
71
|
adantr |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ) → ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) = ( ( nei ‘ 𝐾 ) ‘ { 𝑃 } ) ) |
73 |
3
|
ustuqtop0 |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑁 : 𝑋 ⟶ 𝒫 𝒫 𝑋 ) |
74 |
3
|
ustuqtop1 |
⊢ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ) → 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ) |
75 |
3
|
ustuqtop2 |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) → ( fi ‘ ( 𝑁 ‘ 𝑝 ) ) ⊆ ( 𝑁 ‘ 𝑝 ) ) |
76 |
3
|
ustuqtop3 |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ) → 𝑝 ∈ 𝑎 ) |
77 |
3
|
ustuqtop4 |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ) → ∃ 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ∀ 𝑞 ∈ 𝑏 𝑎 ∈ ( 𝑁 ‘ 𝑞 ) ) |
78 |
3
|
ustuqtop5 |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) → 𝑋 ∈ ( 𝑁 ‘ 𝑝 ) ) |
79 |
2 73 74 75 76 77 78
|
neiptopnei |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑁 = ( 𝑝 ∈ 𝑋 ↦ ( ( nei ‘ 𝐾 ) ‘ { 𝑝 } ) ) ) |
80 |
79
|
adantr |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ) → 𝑁 = ( 𝑝 ∈ 𝑋 ↦ ( ( nei ‘ 𝐾 ) ‘ { 𝑝 } ) ) ) |
81 |
|
simpr |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ) ∧ 𝑝 = 𝑃 ) → 𝑝 = 𝑃 ) |
82 |
81
|
sneqd |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ) ∧ 𝑝 = 𝑃 ) → { 𝑝 } = { 𝑃 } ) |
83 |
82
|
fveq2d |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ) ∧ 𝑝 = 𝑃 ) → ( ( nei ‘ 𝐾 ) ‘ { 𝑝 } ) = ( ( nei ‘ 𝐾 ) ‘ { 𝑃 } ) ) |
84 |
|
simpr |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ) → 𝑃 ∈ 𝑋 ) |
85 |
|
fvexd |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ) → ( ( nei ‘ 𝐾 ) ‘ { 𝑃 } ) ∈ V ) |
86 |
80 83 84 85
|
fvmptd |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ) → ( 𝑁 ‘ 𝑃 ) = ( ( nei ‘ 𝐾 ) ‘ { 𝑃 } ) ) |
87 |
|
mptexg |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ) |
88 |
|
rnexg |
⊢ ( ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V → ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ) |
89 |
87 88
|
syl |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ) |
90 |
89
|
adantr |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ) → ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ) |
91 |
|
nfv |
⊢ Ⅎ 𝑣 𝑃 ∈ 𝑋 |
92 |
|
nfmpt1 |
⊢ Ⅎ 𝑣 ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) |
93 |
92
|
nfrn |
⊢ Ⅎ 𝑣 ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) |
94 |
93
|
nfel1 |
⊢ Ⅎ 𝑣 ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V |
95 |
91 94
|
nfan |
⊢ Ⅎ 𝑣 ( 𝑃 ∈ 𝑋 ∧ ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ) |
96 |
|
nfv |
⊢ Ⅎ 𝑣 𝑝 = 𝑃 |
97 |
95 96
|
nfan |
⊢ Ⅎ 𝑣 ( ( 𝑃 ∈ 𝑋 ∧ ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ) ∧ 𝑝 = 𝑃 ) |
98 |
|
simpr2 |
⊢ ( ( 𝑃 ∈ 𝑋 ∧ ( ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ∧ 𝑝 = 𝑃 ∧ 𝑣 ∈ 𝑈 ) ) → 𝑝 = 𝑃 ) |
99 |
98
|
sneqd |
⊢ ( ( 𝑃 ∈ 𝑋 ∧ ( ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ∧ 𝑝 = 𝑃 ∧ 𝑣 ∈ 𝑈 ) ) → { 𝑝 } = { 𝑃 } ) |
100 |
99
|
imaeq2d |
⊢ ( ( 𝑃 ∈ 𝑋 ∧ ( ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ∧ 𝑝 = 𝑃 ∧ 𝑣 ∈ 𝑈 ) ) → ( 𝑣 “ { 𝑝 } ) = ( 𝑣 “ { 𝑃 } ) ) |
101 |
100
|
3anassrs |
⊢ ( ( ( ( 𝑃 ∈ 𝑋 ∧ ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ) ∧ 𝑝 = 𝑃 ) ∧ 𝑣 ∈ 𝑈 ) → ( 𝑣 “ { 𝑝 } ) = ( 𝑣 “ { 𝑃 } ) ) |
102 |
97 101
|
mpteq2da |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ) ∧ 𝑝 = 𝑃 ) → ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) = ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ) |
103 |
102
|
rneqd |
⊢ ( ( ( 𝑃 ∈ 𝑋 ∧ ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ) ∧ 𝑝 = 𝑃 ) → ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) = ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ) |
104 |
|
simpl |
⊢ ( ( 𝑃 ∈ 𝑋 ∧ ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ) → 𝑃 ∈ 𝑋 ) |
105 |
|
simpr |
⊢ ( ( 𝑃 ∈ 𝑋 ∧ ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ) → ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ) |
106 |
3 103 104 105
|
fvmptd2 |
⊢ ( ( 𝑃 ∈ 𝑋 ∧ ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ∈ V ) → ( 𝑁 ‘ 𝑃 ) = ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ) |
107 |
84 90 106
|
syl2anc |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ) → ( 𝑁 ‘ 𝑃 ) = ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ) |
108 |
72 86 107
|
3eqtr2d |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑃 ∈ 𝑋 ) → ( ( nei ‘ 𝐽 ) ‘ { 𝑃 } ) = ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑃 } ) ) ) |