Step |
Hyp |
Ref |
Expression |
1 |
|
utopustuq.1 |
⊢ 𝑁 = ( 𝑝 ∈ 𝑋 ↦ ran ( 𝑣 ∈ 𝑈 ↦ ( 𝑣 “ { 𝑝 } ) ) ) |
2 |
|
simplll |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) → ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ) |
3 |
|
simplr |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) → 𝑢 ∈ 𝑈 ) |
4 |
|
eqid |
⊢ ( 𝑢 “ { 𝑝 } ) = ( 𝑢 “ { 𝑝 } ) |
5 |
|
imaeq1 |
⊢ ( 𝑤 = 𝑢 → ( 𝑤 “ { 𝑝 } ) = ( 𝑢 “ { 𝑝 } ) ) |
6 |
5
|
rspceeqv |
⊢ ( ( 𝑢 ∈ 𝑈 ∧ ( 𝑢 “ { 𝑝 } ) = ( 𝑢 “ { 𝑝 } ) ) → ∃ 𝑤 ∈ 𝑈 ( 𝑢 “ { 𝑝 } ) = ( 𝑤 “ { 𝑝 } ) ) |
7 |
4 6
|
mpan2 |
⊢ ( 𝑢 ∈ 𝑈 → ∃ 𝑤 ∈ 𝑈 ( 𝑢 “ { 𝑝 } ) = ( 𝑤 “ { 𝑝 } ) ) |
8 |
7
|
adantl |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑢 ∈ 𝑈 ) → ∃ 𝑤 ∈ 𝑈 ( 𝑢 “ { 𝑝 } ) = ( 𝑤 “ { 𝑝 } ) ) |
9 |
|
imaexg |
⊢ ( 𝑢 ∈ 𝑈 → ( 𝑢 “ { 𝑝 } ) ∈ V ) |
10 |
1
|
ustuqtoplem |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑝 } ) ∈ V ) → ( ( 𝑢 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑝 ) ↔ ∃ 𝑤 ∈ 𝑈 ( 𝑢 “ { 𝑝 } ) = ( 𝑤 “ { 𝑝 } ) ) ) |
11 |
9 10
|
sylan2 |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑢 ∈ 𝑈 ) → ( ( 𝑢 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑝 ) ↔ ∃ 𝑤 ∈ 𝑈 ( 𝑢 “ { 𝑝 } ) = ( 𝑤 “ { 𝑝 } ) ) ) |
12 |
8 11
|
mpbird |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑢 ∈ 𝑈 ) → ( 𝑢 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑝 ) ) |
13 |
2 3 12
|
syl2anc |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) → ( 𝑢 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑝 ) ) |
14 |
|
simp-5l |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) → 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ) |
15 |
2
|
simpld |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) → 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ) |
16 |
|
simp-4r |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) → 𝑝 ∈ 𝑋 ) |
17 |
|
ustimasn |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑢 ∈ 𝑈 ∧ 𝑝 ∈ 𝑋 ) → ( 𝑢 “ { 𝑝 } ) ⊆ 𝑋 ) |
18 |
15 3 16 17
|
syl3anc |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) → ( 𝑢 “ { 𝑝 } ) ⊆ 𝑋 ) |
19 |
18
|
sselda |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) → 𝑞 ∈ 𝑋 ) |
20 |
14 19
|
jca |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) → ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ) |
21 |
|
simplr |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) |
22 |
|
simp-6l |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ) |
23 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → 𝑢 ∈ 𝑈 ) |
24 |
|
ustrel |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑢 ∈ 𝑈 ) → Rel 𝑢 ) |
25 |
22 23 24
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → Rel 𝑢 ) |
26 |
|
elrelimasn |
⊢ ( Rel 𝑢 → ( 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ↔ 𝑝 𝑢 𝑞 ) ) |
27 |
25 26
|
syl |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → ( 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ↔ 𝑝 𝑢 𝑞 ) ) |
28 |
21 27
|
mpbid |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → 𝑝 𝑢 𝑞 ) |
29 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) |
30 |
|
elrelimasn |
⊢ ( Rel 𝑢 → ( 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ↔ 𝑞 𝑢 𝑟 ) ) |
31 |
25 30
|
syl |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → ( 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ↔ 𝑞 𝑢 𝑟 ) ) |
32 |
29 31
|
mpbid |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → 𝑞 𝑢 𝑟 ) |
33 |
|
vex |
⊢ 𝑝 ∈ V |
34 |
|
vex |
⊢ 𝑟 ∈ V |
35 |
33 34
|
brco |
⊢ ( 𝑝 ( 𝑢 ∘ 𝑢 ) 𝑟 ↔ ∃ 𝑞 ( 𝑝 𝑢 𝑞 ∧ 𝑞 𝑢 𝑟 ) ) |
36 |
35
|
biimpri |
⊢ ( ∃ 𝑞 ( 𝑝 𝑢 𝑞 ∧ 𝑞 𝑢 𝑟 ) → 𝑝 ( 𝑢 ∘ 𝑢 ) 𝑟 ) |
37 |
36
|
19.23bi |
⊢ ( ( 𝑝 𝑢 𝑞 ∧ 𝑞 𝑢 𝑟 ) → 𝑝 ( 𝑢 ∘ 𝑢 ) 𝑟 ) |
38 |
28 32 37
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → 𝑝 ( 𝑢 ∘ 𝑢 ) 𝑟 ) |
39 |
|
simpllr |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) |
40 |
39
|
ssbrd |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → ( 𝑝 ( 𝑢 ∘ 𝑢 ) 𝑟 → 𝑝 𝑤 𝑟 ) ) |
41 |
38 40
|
mpd |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → 𝑝 𝑤 𝑟 ) |
42 |
|
simp-5r |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → 𝑤 ∈ 𝑈 ) |
43 |
|
ustrel |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) → Rel 𝑤 ) |
44 |
22 42 43
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → Rel 𝑤 ) |
45 |
|
elrelimasn |
⊢ ( Rel 𝑤 → ( 𝑟 ∈ ( 𝑤 “ { 𝑝 } ) ↔ 𝑝 𝑤 𝑟 ) ) |
46 |
44 45
|
syl |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → ( 𝑟 ∈ ( 𝑤 “ { 𝑝 } ) ↔ 𝑝 𝑤 𝑟 ) ) |
47 |
41 46
|
mpbird |
⊢ ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) ∧ 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) ) → 𝑟 ∈ ( 𝑤 “ { 𝑝 } ) ) |
48 |
47
|
ex |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) → ( 𝑟 ∈ ( 𝑢 “ { 𝑞 } ) → 𝑟 ∈ ( 𝑤 “ { 𝑝 } ) ) ) |
49 |
48
|
ssrdv |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) → ( 𝑢 “ { 𝑞 } ) ⊆ ( 𝑤 “ { 𝑝 } ) ) |
50 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) → 𝑤 ∈ 𝑈 ) |
51 |
16
|
adantr |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) → 𝑝 ∈ 𝑋 ) |
52 |
|
ustimasn |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ∧ 𝑝 ∈ 𝑋 ) → ( 𝑤 “ { 𝑝 } ) ⊆ 𝑋 ) |
53 |
14 50 51 52
|
syl3anc |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) → ( 𝑤 “ { 𝑝 } ) ⊆ 𝑋 ) |
54 |
20 49 53
|
3jca |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) → ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ ( 𝑤 “ { 𝑝 } ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑋 ) ) |
55 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) → 𝑢 ∈ 𝑈 ) |
56 |
|
eqidd |
⊢ ( 𝑢 ∈ 𝑈 → ( 𝑢 “ { 𝑞 } ) = ( 𝑢 “ { 𝑞 } ) ) |
57 |
|
imaeq1 |
⊢ ( 𝑤 = 𝑢 → ( 𝑤 “ { 𝑞 } ) = ( 𝑢 “ { 𝑞 } ) ) |
58 |
57
|
rspceeqv |
⊢ ( ( 𝑢 ∈ 𝑈 ∧ ( 𝑢 “ { 𝑞 } ) = ( 𝑢 “ { 𝑞 } ) ) → ∃ 𝑤 ∈ 𝑈 ( 𝑢 “ { 𝑞 } ) = ( 𝑤 “ { 𝑞 } ) ) |
59 |
56 58
|
mpdan |
⊢ ( 𝑢 ∈ 𝑈 → ∃ 𝑤 ∈ 𝑈 ( 𝑢 “ { 𝑞 } ) = ( 𝑤 “ { 𝑞 } ) ) |
60 |
59
|
adantl |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑢 ∈ 𝑈 ) → ∃ 𝑤 ∈ 𝑈 ( 𝑢 “ { 𝑞 } ) = ( 𝑤 “ { 𝑞 } ) ) |
61 |
|
imaexg |
⊢ ( 𝑢 ∈ 𝑈 → ( 𝑢 “ { 𝑞 } ) ∈ V ) |
62 |
1
|
ustuqtoplem |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ V ) → ( ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ↔ ∃ 𝑤 ∈ 𝑈 ( 𝑢 “ { 𝑞 } ) = ( 𝑤 “ { 𝑞 } ) ) ) |
63 |
61 62
|
sylan2 |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑢 ∈ 𝑈 ) → ( ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ↔ ∃ 𝑤 ∈ 𝑈 ( 𝑢 “ { 𝑞 } ) = ( 𝑤 “ { 𝑞 } ) ) ) |
64 |
60 63
|
mpbird |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑢 ∈ 𝑈 ) → ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) |
65 |
14 19 55 64
|
syl21anc |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) → ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) |
66 |
54 65
|
jca |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) → ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ ( 𝑤 “ { 𝑝 } ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ) |
67 |
|
imaexg |
⊢ ( 𝑤 ∈ 𝑈 → ( 𝑤 “ { 𝑝 } ) ∈ V ) |
68 |
|
sseq2 |
⊢ ( 𝑏 = ( 𝑤 “ { 𝑝 } ) → ( ( 𝑢 “ { 𝑞 } ) ⊆ 𝑏 ↔ ( 𝑢 “ { 𝑞 } ) ⊆ ( 𝑤 “ { 𝑝 } ) ) ) |
69 |
|
sseq1 |
⊢ ( 𝑏 = ( 𝑤 “ { 𝑝 } ) → ( 𝑏 ⊆ 𝑋 ↔ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑋 ) ) |
70 |
68 69
|
3anbi23d |
⊢ ( 𝑏 = ( 𝑤 “ { 𝑝 } ) → ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ↔ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ ( 𝑤 “ { 𝑝 } ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑋 ) ) ) |
71 |
70
|
anbi1d |
⊢ ( 𝑏 = ( 𝑤 “ { 𝑝 } ) → ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ↔ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ ( 𝑤 “ { 𝑝 } ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ) ) |
72 |
71
|
anbi1d |
⊢ ( 𝑏 = ( 𝑤 “ { 𝑝 } ) → ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ∧ 𝑢 ∈ 𝑈 ) ↔ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ ( 𝑤 “ { 𝑝 } ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ∧ 𝑢 ∈ 𝑈 ) ) ) |
73 |
|
eleq1 |
⊢ ( 𝑏 = ( 𝑤 “ { 𝑝 } ) → ( 𝑏 ∈ ( 𝑁 ‘ 𝑞 ) ↔ ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ) |
74 |
72 73
|
imbi12d |
⊢ ( 𝑏 = ( 𝑤 “ { 𝑝 } ) → ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ∧ 𝑢 ∈ 𝑈 ) → 𝑏 ∈ ( 𝑁 ‘ 𝑞 ) ) ↔ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ ( 𝑤 “ { 𝑝 } ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ∧ 𝑢 ∈ 𝑈 ) → ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ) ) |
75 |
|
sseq1 |
⊢ ( 𝑎 = ( 𝑢 “ { 𝑞 } ) → ( 𝑎 ⊆ 𝑏 ↔ ( 𝑢 “ { 𝑞 } ) ⊆ 𝑏 ) ) |
76 |
75
|
3anbi2d |
⊢ ( 𝑎 = ( 𝑢 “ { 𝑞 } ) → ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ↔ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ) ) |
77 |
|
eleq1 |
⊢ ( 𝑎 = ( 𝑢 “ { 𝑞 } ) → ( 𝑎 ∈ ( 𝑁 ‘ 𝑞 ) ↔ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ) |
78 |
76 77
|
anbi12d |
⊢ ( 𝑎 = ( 𝑢 “ { 𝑞 } ) → ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ 𝑎 ∈ ( 𝑁 ‘ 𝑞 ) ) ↔ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ) ) |
79 |
78
|
imbi1d |
⊢ ( 𝑎 = ( 𝑢 “ { 𝑞 } ) → ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ 𝑎 ∈ ( 𝑁 ‘ 𝑞 ) ) → 𝑏 ∈ ( 𝑁 ‘ 𝑞 ) ) ↔ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) → 𝑏 ∈ ( 𝑁 ‘ 𝑞 ) ) ) ) |
80 |
|
eleq1 |
⊢ ( 𝑝 = 𝑞 → ( 𝑝 ∈ 𝑋 ↔ 𝑞 ∈ 𝑋 ) ) |
81 |
80
|
anbi2d |
⊢ ( 𝑝 = 𝑞 → ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ↔ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ) ) |
82 |
81
|
3anbi1d |
⊢ ( 𝑝 = 𝑞 → ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ↔ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ) ) |
83 |
|
fveq2 |
⊢ ( 𝑝 = 𝑞 → ( 𝑁 ‘ 𝑝 ) = ( 𝑁 ‘ 𝑞 ) ) |
84 |
83
|
eleq2d |
⊢ ( 𝑝 = 𝑞 → ( 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ↔ 𝑎 ∈ ( 𝑁 ‘ 𝑞 ) ) ) |
85 |
82 84
|
anbi12d |
⊢ ( 𝑝 = 𝑞 → ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ) ↔ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ 𝑎 ∈ ( 𝑁 ‘ 𝑞 ) ) ) ) |
86 |
83
|
eleq2d |
⊢ ( 𝑝 = 𝑞 → ( 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ↔ 𝑏 ∈ ( 𝑁 ‘ 𝑞 ) ) ) |
87 |
85 86
|
imbi12d |
⊢ ( 𝑝 = 𝑞 → ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ) → 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ) ↔ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ 𝑎 ∈ ( 𝑁 ‘ 𝑞 ) ) → 𝑏 ∈ ( 𝑁 ‘ 𝑞 ) ) ) ) |
88 |
1
|
ustuqtop1 |
⊢ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ) → 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ) |
89 |
87 88
|
chvarvv |
⊢ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ 𝑎 ∈ ( 𝑁 ‘ 𝑞 ) ) → 𝑏 ∈ ( 𝑁 ‘ 𝑞 ) ) |
90 |
79 89
|
vtoclg |
⊢ ( ( 𝑢 “ { 𝑞 } ) ∈ V → ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) → 𝑏 ∈ ( 𝑁 ‘ 𝑞 ) ) ) |
91 |
61 90
|
syl |
⊢ ( 𝑢 ∈ 𝑈 → ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) → 𝑏 ∈ ( 𝑁 ‘ 𝑞 ) ) ) |
92 |
91
|
impcom |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ∧ 𝑢 ∈ 𝑈 ) → 𝑏 ∈ ( 𝑁 ‘ 𝑞 ) ) |
93 |
74 92
|
vtoclg |
⊢ ( ( 𝑤 “ { 𝑝 } ) ∈ V → ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ ( 𝑤 “ { 𝑝 } ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ∧ 𝑢 ∈ 𝑈 ) → ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ) |
94 |
67 93
|
syl |
⊢ ( 𝑤 ∈ 𝑈 → ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ ( 𝑤 “ { 𝑝 } ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ∧ 𝑢 ∈ 𝑈 ) → ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ) |
95 |
94
|
impcom |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑞 ∈ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ⊆ ( 𝑤 “ { 𝑝 } ) ∧ ( 𝑤 “ { 𝑝 } ) ⊆ 𝑋 ) ∧ ( 𝑢 “ { 𝑞 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑤 ∈ 𝑈 ) → ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) |
96 |
66 55 50 95
|
syl21anc |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) ∧ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ) → ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) |
97 |
96
|
ralrimiva |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) → ∀ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) |
98 |
|
raleq |
⊢ ( 𝑏 = ( 𝑢 “ { 𝑝 } ) → ( ∀ 𝑞 ∈ 𝑏 ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ↔ ∀ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ) |
99 |
98
|
rspcev |
⊢ ( ( ( 𝑢 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑝 ) ∧ ∀ 𝑞 ∈ ( 𝑢 “ { 𝑝 } ) ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) → ∃ 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ∀ 𝑞 ∈ 𝑏 ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) |
100 |
13 97 99
|
syl2anc |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 ∈ 𝑈 ) ∧ ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) → ∃ 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ∀ 𝑞 ∈ 𝑏 ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) |
101 |
|
ustexhalf |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) → ∃ 𝑢 ∈ 𝑈 ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) |
102 |
101
|
adantlr |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) → ∃ 𝑢 ∈ 𝑈 ( 𝑢 ∘ 𝑢 ) ⊆ 𝑤 ) |
103 |
100 102
|
r19.29a |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) → ∃ 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ∀ 𝑞 ∈ 𝑏 ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) |
104 |
103
|
adantr |
⊢ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑎 = ( 𝑤 “ { 𝑝 } ) ) → ∃ 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ∀ 𝑞 ∈ 𝑏 ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) |
105 |
|
eleq1 |
⊢ ( 𝑎 = ( 𝑤 “ { 𝑝 } ) → ( 𝑎 ∈ ( 𝑁 ‘ 𝑞 ) ↔ ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ) |
106 |
105
|
rexralbidv |
⊢ ( 𝑎 = ( 𝑤 “ { 𝑝 } ) → ( ∃ 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ∀ 𝑞 ∈ 𝑏 𝑎 ∈ ( 𝑁 ‘ 𝑞 ) ↔ ∃ 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ∀ 𝑞 ∈ 𝑏 ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ) |
107 |
106
|
adantl |
⊢ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑎 = ( 𝑤 “ { 𝑝 } ) ) → ( ∃ 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ∀ 𝑞 ∈ 𝑏 𝑎 ∈ ( 𝑁 ‘ 𝑞 ) ↔ ∃ 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ∀ 𝑞 ∈ 𝑏 ( 𝑤 “ { 𝑝 } ) ∈ ( 𝑁 ‘ 𝑞 ) ) ) |
108 |
104 107
|
mpbird |
⊢ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑎 = ( 𝑤 “ { 𝑝 } ) ) → ∃ 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ∀ 𝑞 ∈ 𝑏 𝑎 ∈ ( 𝑁 ‘ 𝑞 ) ) |
109 |
108
|
adantllr |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑎 = ( 𝑤 “ { 𝑝 } ) ) → ∃ 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ∀ 𝑞 ∈ 𝑏 𝑎 ∈ ( 𝑁 ‘ 𝑞 ) ) |
110 |
|
vex |
⊢ 𝑎 ∈ V |
111 |
1
|
ustuqtoplem |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑎 ∈ V ) → ( 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ↔ ∃ 𝑤 ∈ 𝑈 𝑎 = ( 𝑤 “ { 𝑝 } ) ) ) |
112 |
110 111
|
mpan2 |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) → ( 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ↔ ∃ 𝑤 ∈ 𝑈 𝑎 = ( 𝑤 “ { 𝑝 } ) ) ) |
113 |
112
|
biimpa |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ) → ∃ 𝑤 ∈ 𝑈 𝑎 = ( 𝑤 “ { 𝑝 } ) ) |
114 |
109 113
|
r19.29a |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ) ∧ 𝑎 ∈ ( 𝑁 ‘ 𝑝 ) ) → ∃ 𝑏 ∈ ( 𝑁 ‘ 𝑝 ) ∀ 𝑞 ∈ 𝑏 𝑎 ∈ ( 𝑁 ‘ 𝑞 ) ) |