Step |
Hyp |
Ref |
Expression |
1 |
|
cnprest.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
|
eqid |
⊢ ∪ 𝐾 = ∪ 𝐾 |
3 |
1 2
|
cnpf |
⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝐹 : 𝑋 ⟶ ∪ 𝐾 ) |
4 |
3
|
3ad2ant3 |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝐹 : 𝑋 ⟶ ∪ 𝐾 ) |
5 |
|
simp1 |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝐴 ⊆ 𝑋 ) |
6 |
4 5
|
fssresd |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ( 𝐹 ↾ 𝐴 ) : 𝐴 ⟶ ∪ 𝐾 ) |
7 |
|
simpl2 |
⊢ ( ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) ∧ 𝑦 ∈ 𝐾 ) → 𝑃 ∈ 𝐴 ) |
8 |
7
|
fvresd |
⊢ ( ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) ∧ 𝑦 ∈ 𝐾 ) → ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) |
9 |
8
|
eleq1d |
⊢ ( ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) ∧ 𝑦 ∈ 𝐾 ) → ( ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑃 ) ∈ 𝑦 ↔ ( 𝐹 ‘ 𝑃 ) ∈ 𝑦 ) ) |
10 |
|
cnpimaex |
⊢ ( ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ∧ 𝑦 ∈ 𝐾 ∧ ( 𝐹 ‘ 𝑃 ) ∈ 𝑦 ) → ∃ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 ∧ ( 𝐹 “ 𝑥 ) ⊆ 𝑦 ) ) |
11 |
10
|
3expia |
⊢ ( ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ∧ 𝑦 ∈ 𝐾 ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝑦 → ∃ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 ∧ ( 𝐹 “ 𝑥 ) ⊆ 𝑦 ) ) ) |
12 |
11
|
3ad2antl3 |
⊢ ( ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) ∧ 𝑦 ∈ 𝐾 ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝑦 → ∃ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 ∧ ( 𝐹 “ 𝑥 ) ⊆ 𝑦 ) ) ) |
13 |
|
idd |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ( 𝑃 ∈ 𝑥 → 𝑃 ∈ 𝑥 ) ) |
14 |
|
simp2 |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝑃 ∈ 𝐴 ) |
15 |
13 14
|
jctird |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ( 𝑃 ∈ 𝑥 → ( 𝑃 ∈ 𝑥 ∧ 𝑃 ∈ 𝐴 ) ) ) |
16 |
|
elin |
⊢ ( 𝑃 ∈ ( 𝑥 ∩ 𝐴 ) ↔ ( 𝑃 ∈ 𝑥 ∧ 𝑃 ∈ 𝐴 ) ) |
17 |
15 16
|
syl6ibr |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ( 𝑃 ∈ 𝑥 → 𝑃 ∈ ( 𝑥 ∩ 𝐴 ) ) ) |
18 |
|
inss1 |
⊢ ( 𝑥 ∩ 𝐴 ) ⊆ 𝑥 |
19 |
|
imass2 |
⊢ ( ( 𝑥 ∩ 𝐴 ) ⊆ 𝑥 → ( 𝐹 “ ( 𝑥 ∩ 𝐴 ) ) ⊆ ( 𝐹 “ 𝑥 ) ) |
20 |
18 19
|
ax-mp |
⊢ ( 𝐹 “ ( 𝑥 ∩ 𝐴 ) ) ⊆ ( 𝐹 “ 𝑥 ) |
21 |
|
id |
⊢ ( ( 𝐹 “ 𝑥 ) ⊆ 𝑦 → ( 𝐹 “ 𝑥 ) ⊆ 𝑦 ) |
22 |
20 21
|
sstrid |
⊢ ( ( 𝐹 “ 𝑥 ) ⊆ 𝑦 → ( 𝐹 “ ( 𝑥 ∩ 𝐴 ) ) ⊆ 𝑦 ) |
23 |
17 22
|
anim12d1 |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ( ( 𝑃 ∈ 𝑥 ∧ ( 𝐹 “ 𝑥 ) ⊆ 𝑦 ) → ( 𝑃 ∈ ( 𝑥 ∩ 𝐴 ) ∧ ( 𝐹 “ ( 𝑥 ∩ 𝐴 ) ) ⊆ 𝑦 ) ) ) |
24 |
23
|
reximdv |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ( ∃ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 ∧ ( 𝐹 “ 𝑥 ) ⊆ 𝑦 ) → ∃ 𝑥 ∈ 𝐽 ( 𝑃 ∈ ( 𝑥 ∩ 𝐴 ) ∧ ( 𝐹 “ ( 𝑥 ∩ 𝐴 ) ) ⊆ 𝑦 ) ) ) |
25 |
|
vex |
⊢ 𝑥 ∈ V |
26 |
25
|
inex1 |
⊢ ( 𝑥 ∩ 𝐴 ) ∈ V |
27 |
26
|
a1i |
⊢ ( ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) ∧ 𝑥 ∈ 𝐽 ) → ( 𝑥 ∩ 𝐴 ) ∈ V ) |
28 |
|
cnptop1 |
⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝐽 ∈ Top ) |
29 |
28
|
3ad2ant3 |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝐽 ∈ Top ) |
30 |
29
|
uniexd |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ∪ 𝐽 ∈ V ) |
31 |
5 1
|
sseqtrdi |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝐴 ⊆ ∪ 𝐽 ) |
32 |
30 31
|
ssexd |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝐴 ∈ V ) |
33 |
|
elrest |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ V ) → ( 𝑧 ∈ ( 𝐽 ↾t 𝐴 ) ↔ ∃ 𝑥 ∈ 𝐽 𝑧 = ( 𝑥 ∩ 𝐴 ) ) ) |
34 |
29 32 33
|
syl2anc |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ( 𝑧 ∈ ( 𝐽 ↾t 𝐴 ) ↔ ∃ 𝑥 ∈ 𝐽 𝑧 = ( 𝑥 ∩ 𝐴 ) ) ) |
35 |
|
simpr |
⊢ ( ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) ∧ 𝑧 = ( 𝑥 ∩ 𝐴 ) ) → 𝑧 = ( 𝑥 ∩ 𝐴 ) ) |
36 |
35
|
eleq2d |
⊢ ( ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) ∧ 𝑧 = ( 𝑥 ∩ 𝐴 ) ) → ( 𝑃 ∈ 𝑧 ↔ 𝑃 ∈ ( 𝑥 ∩ 𝐴 ) ) ) |
37 |
35
|
imaeq2d |
⊢ ( ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) ∧ 𝑧 = ( 𝑥 ∩ 𝐴 ) ) → ( ( 𝐹 ↾ 𝐴 ) “ 𝑧 ) = ( ( 𝐹 ↾ 𝐴 ) “ ( 𝑥 ∩ 𝐴 ) ) ) |
38 |
|
inss2 |
⊢ ( 𝑥 ∩ 𝐴 ) ⊆ 𝐴 |
39 |
|
resima2 |
⊢ ( ( 𝑥 ∩ 𝐴 ) ⊆ 𝐴 → ( ( 𝐹 ↾ 𝐴 ) “ ( 𝑥 ∩ 𝐴 ) ) = ( 𝐹 “ ( 𝑥 ∩ 𝐴 ) ) ) |
40 |
38 39
|
ax-mp |
⊢ ( ( 𝐹 ↾ 𝐴 ) “ ( 𝑥 ∩ 𝐴 ) ) = ( 𝐹 “ ( 𝑥 ∩ 𝐴 ) ) |
41 |
37 40
|
eqtrdi |
⊢ ( ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) ∧ 𝑧 = ( 𝑥 ∩ 𝐴 ) ) → ( ( 𝐹 ↾ 𝐴 ) “ 𝑧 ) = ( 𝐹 “ ( 𝑥 ∩ 𝐴 ) ) ) |
42 |
41
|
sseq1d |
⊢ ( ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) ∧ 𝑧 = ( 𝑥 ∩ 𝐴 ) ) → ( ( ( 𝐹 ↾ 𝐴 ) “ 𝑧 ) ⊆ 𝑦 ↔ ( 𝐹 “ ( 𝑥 ∩ 𝐴 ) ) ⊆ 𝑦 ) ) |
43 |
36 42
|
anbi12d |
⊢ ( ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) ∧ 𝑧 = ( 𝑥 ∩ 𝐴 ) ) → ( ( 𝑃 ∈ 𝑧 ∧ ( ( 𝐹 ↾ 𝐴 ) “ 𝑧 ) ⊆ 𝑦 ) ↔ ( 𝑃 ∈ ( 𝑥 ∩ 𝐴 ) ∧ ( 𝐹 “ ( 𝑥 ∩ 𝐴 ) ) ⊆ 𝑦 ) ) ) |
44 |
27 34 43
|
rexxfr2d |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ( ∃ 𝑧 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝑃 ∈ 𝑧 ∧ ( ( 𝐹 ↾ 𝐴 ) “ 𝑧 ) ⊆ 𝑦 ) ↔ ∃ 𝑥 ∈ 𝐽 ( 𝑃 ∈ ( 𝑥 ∩ 𝐴 ) ∧ ( 𝐹 “ ( 𝑥 ∩ 𝐴 ) ) ⊆ 𝑦 ) ) ) |
45 |
24 44
|
sylibrd |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ( ∃ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 ∧ ( 𝐹 “ 𝑥 ) ⊆ 𝑦 ) → ∃ 𝑧 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝑃 ∈ 𝑧 ∧ ( ( 𝐹 ↾ 𝐴 ) “ 𝑧 ) ⊆ 𝑦 ) ) ) |
46 |
45
|
adantr |
⊢ ( ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) ∧ 𝑦 ∈ 𝐾 ) → ( ∃ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 ∧ ( 𝐹 “ 𝑥 ) ⊆ 𝑦 ) → ∃ 𝑧 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝑃 ∈ 𝑧 ∧ ( ( 𝐹 ↾ 𝐴 ) “ 𝑧 ) ⊆ 𝑦 ) ) ) |
47 |
12 46
|
syld |
⊢ ( ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) ∧ 𝑦 ∈ 𝐾 ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝑦 → ∃ 𝑧 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝑃 ∈ 𝑧 ∧ ( ( 𝐹 ↾ 𝐴 ) “ 𝑧 ) ⊆ 𝑦 ) ) ) |
48 |
9 47
|
sylbid |
⊢ ( ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) ∧ 𝑦 ∈ 𝐾 ) → ( ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑃 ) ∈ 𝑦 → ∃ 𝑧 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝑃 ∈ 𝑧 ∧ ( ( 𝐹 ↾ 𝐴 ) “ 𝑧 ) ⊆ 𝑦 ) ) ) |
49 |
48
|
ralrimiva |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ∀ 𝑦 ∈ 𝐾 ( ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑃 ) ∈ 𝑦 → ∃ 𝑧 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝑃 ∈ 𝑧 ∧ ( ( 𝐹 ↾ 𝐴 ) “ 𝑧 ) ⊆ 𝑦 ) ) ) |
50 |
1
|
toptopon |
⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
51 |
29 50
|
sylib |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
52 |
|
resttopon |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) → ( 𝐽 ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) ) |
53 |
51 5 52
|
syl2anc |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ( 𝐽 ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) ) |
54 |
|
cnptop2 |
⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝐾 ∈ Top ) |
55 |
54
|
3ad2ant3 |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝐾 ∈ Top ) |
56 |
2
|
toptopon |
⊢ ( 𝐾 ∈ Top ↔ 𝐾 ∈ ( TopOn ‘ ∪ 𝐾 ) ) |
57 |
55 56
|
sylib |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝐾 ∈ ( TopOn ‘ ∪ 𝐾 ) ) |
58 |
|
iscnp |
⊢ ( ( ( 𝐽 ↾t 𝐴 ) ∈ ( TopOn ‘ 𝐴 ) ∧ 𝐾 ∈ ( TopOn ‘ ∪ 𝐾 ) ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝐹 ↾ 𝐴 ) ∈ ( ( ( 𝐽 ↾t 𝐴 ) CnP 𝐾 ) ‘ 𝑃 ) ↔ ( ( 𝐹 ↾ 𝐴 ) : 𝐴 ⟶ ∪ 𝐾 ∧ ∀ 𝑦 ∈ 𝐾 ( ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑃 ) ∈ 𝑦 → ∃ 𝑧 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝑃 ∈ 𝑧 ∧ ( ( 𝐹 ↾ 𝐴 ) “ 𝑧 ) ⊆ 𝑦 ) ) ) ) ) |
59 |
53 57 14 58
|
syl3anc |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ( ( 𝐹 ↾ 𝐴 ) ∈ ( ( ( 𝐽 ↾t 𝐴 ) CnP 𝐾 ) ‘ 𝑃 ) ↔ ( ( 𝐹 ↾ 𝐴 ) : 𝐴 ⟶ ∪ 𝐾 ∧ ∀ 𝑦 ∈ 𝐾 ( ( ( 𝐹 ↾ 𝐴 ) ‘ 𝑃 ) ∈ 𝑦 → ∃ 𝑧 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝑃 ∈ 𝑧 ∧ ( ( 𝐹 ↾ 𝐴 ) “ 𝑧 ) ⊆ 𝑦 ) ) ) ) ) |
60 |
6 49 59
|
mpbir2and |
⊢ ( ( 𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → ( 𝐹 ↾ 𝐴 ) ∈ ( ( ( 𝐽 ↾t 𝐴 ) CnP 𝐾 ) ‘ 𝑃 ) ) |