| Step |
Hyp |
Ref |
Expression |
| 1 |
|
neitr.1 |
⊢ 𝑋 = ∪ 𝐽 |
| 2 |
|
nfv |
⊢ Ⅎ 𝑑 ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) |
| 3 |
|
nfv |
⊢ Ⅎ 𝑑 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) |
| 4 |
|
nfre1 |
⊢ Ⅎ 𝑑 ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) |
| 5 |
3 4
|
nfan |
⊢ Ⅎ 𝑑 ( 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ∧ ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) |
| 6 |
2 5
|
nfan |
⊢ Ⅎ 𝑑 ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ ( 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ∧ ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ) |
| 7 |
|
simpl |
⊢ ( ( 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ∧ ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) → 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) |
| 8 |
7
|
anim2i |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ ( 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ∧ ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ) → ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ) |
| 9 |
|
simp-5r |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) |
| 10 |
|
simp1 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → 𝐽 ∈ Top ) |
| 11 |
|
simp2 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → 𝐴 ⊆ 𝑋 ) |
| 12 |
1
|
restuni |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ) → 𝐴 = ∪ ( 𝐽 ↾t 𝐴 ) ) |
| 13 |
10 11 12
|
syl2anc |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → 𝐴 = ∪ ( 𝐽 ↾t 𝐴 ) ) |
| 14 |
13
|
ad5antr |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → 𝐴 = ∪ ( 𝐽 ↾t 𝐴 ) ) |
| 15 |
9 14
|
sseqtrrd |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → 𝑐 ⊆ 𝐴 ) |
| 16 |
11
|
ad5antr |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → 𝐴 ⊆ 𝑋 ) |
| 17 |
15 16
|
sstrd |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → 𝑐 ⊆ 𝑋 ) |
| 18 |
10
|
ad5antr |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → 𝐽 ∈ Top ) |
| 19 |
|
simplr |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → 𝑒 ∈ 𝐽 ) |
| 20 |
1
|
eltopss |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑒 ∈ 𝐽 ) → 𝑒 ⊆ 𝑋 ) |
| 21 |
18 19 20
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → 𝑒 ⊆ 𝑋 ) |
| 22 |
21
|
ssdifssd |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → ( 𝑒 ∖ 𝐴 ) ⊆ 𝑋 ) |
| 23 |
17 22
|
unssd |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ⊆ 𝑋 ) |
| 24 |
|
simpr1l |
⊢ ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ∧ 𝑒 ∈ 𝐽 ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) ) → 𝐵 ⊆ 𝑑 ) |
| 25 |
24
|
3anassrs |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → 𝐵 ⊆ 𝑑 ) |
| 26 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → 𝑑 = ( 𝑒 ∩ 𝐴 ) ) |
| 27 |
25 26
|
sseqtrd |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → 𝐵 ⊆ ( 𝑒 ∩ 𝐴 ) ) |
| 28 |
|
inss1 |
⊢ ( 𝑒 ∩ 𝐴 ) ⊆ 𝑒 |
| 29 |
27 28
|
sstrdi |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → 𝐵 ⊆ 𝑒 ) |
| 30 |
|
inundif |
⊢ ( ( 𝑒 ∩ 𝐴 ) ∪ ( 𝑒 ∖ 𝐴 ) ) = 𝑒 |
| 31 |
|
simpr1r |
⊢ ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ∧ 𝑒 ∈ 𝐽 ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) ) → 𝑑 ⊆ 𝑐 ) |
| 32 |
31
|
3anassrs |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → 𝑑 ⊆ 𝑐 ) |
| 33 |
26 32
|
eqsstrrd |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → ( 𝑒 ∩ 𝐴 ) ⊆ 𝑐 ) |
| 34 |
|
unss1 |
⊢ ( ( 𝑒 ∩ 𝐴 ) ⊆ 𝑐 → ( ( 𝑒 ∩ 𝐴 ) ∪ ( 𝑒 ∖ 𝐴 ) ) ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ) |
| 35 |
33 34
|
syl |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → ( ( 𝑒 ∩ 𝐴 ) ∪ ( 𝑒 ∖ 𝐴 ) ) ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ) |
| 36 |
30 35
|
eqsstrrid |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → 𝑒 ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ) |
| 37 |
|
sseq2 |
⊢ ( 𝑏 = 𝑒 → ( 𝐵 ⊆ 𝑏 ↔ 𝐵 ⊆ 𝑒 ) ) |
| 38 |
|
sseq1 |
⊢ ( 𝑏 = 𝑒 → ( 𝑏 ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ↔ 𝑒 ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ) ) |
| 39 |
37 38
|
anbi12d |
⊢ ( 𝑏 = 𝑒 → ( ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ) ↔ ( 𝐵 ⊆ 𝑒 ∧ 𝑒 ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ) ) ) |
| 40 |
39
|
rspcev |
⊢ ( ( 𝑒 ∈ 𝐽 ∧ ( 𝐵 ⊆ 𝑒 ∧ 𝑒 ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ) ) → ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ) ) |
| 41 |
19 29 36 40
|
syl12anc |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ) ) |
| 42 |
|
indir |
⊢ ( ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ∩ 𝐴 ) = ( ( 𝑐 ∩ 𝐴 ) ∪ ( ( 𝑒 ∖ 𝐴 ) ∩ 𝐴 ) ) |
| 43 |
|
disjdifr |
⊢ ( ( 𝑒 ∖ 𝐴 ) ∩ 𝐴 ) = ∅ |
| 44 |
43
|
uneq2i |
⊢ ( ( 𝑐 ∩ 𝐴 ) ∪ ( ( 𝑒 ∖ 𝐴 ) ∩ 𝐴 ) ) = ( ( 𝑐 ∩ 𝐴 ) ∪ ∅ ) |
| 45 |
|
un0 |
⊢ ( ( 𝑐 ∩ 𝐴 ) ∪ ∅ ) = ( 𝑐 ∩ 𝐴 ) |
| 46 |
42 44 45
|
3eqtri |
⊢ ( ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ∩ 𝐴 ) = ( 𝑐 ∩ 𝐴 ) |
| 47 |
|
df-ss |
⊢ ( 𝑐 ⊆ 𝐴 ↔ ( 𝑐 ∩ 𝐴 ) = 𝑐 ) |
| 48 |
47
|
biimpi |
⊢ ( 𝑐 ⊆ 𝐴 → ( 𝑐 ∩ 𝐴 ) = 𝑐 ) |
| 49 |
46 48
|
eqtr2id |
⊢ ( 𝑐 ⊆ 𝐴 → 𝑐 = ( ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ∩ 𝐴 ) ) |
| 50 |
15 49
|
syl |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → 𝑐 = ( ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ∩ 𝐴 ) ) |
| 51 |
|
vex |
⊢ 𝑐 ∈ V |
| 52 |
|
vex |
⊢ 𝑒 ∈ V |
| 53 |
52
|
difexi |
⊢ ( 𝑒 ∖ 𝐴 ) ∈ V |
| 54 |
51 53
|
unex |
⊢ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ∈ V |
| 55 |
|
sseq1 |
⊢ ( 𝑎 = ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) → ( 𝑎 ⊆ 𝑋 ↔ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ⊆ 𝑋 ) ) |
| 56 |
|
sseq2 |
⊢ ( 𝑎 = ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) → ( 𝑏 ⊆ 𝑎 ↔ 𝑏 ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ) ) |
| 57 |
56
|
anbi2d |
⊢ ( 𝑎 = ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) → ( ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ↔ ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ) ) ) |
| 58 |
57
|
rexbidv |
⊢ ( 𝑎 = ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) → ( ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ↔ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ) ) ) |
| 59 |
55 58
|
anbi12d |
⊢ ( 𝑎 = ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) → ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ↔ ( ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ) ) ) ) |
| 60 |
|
ineq1 |
⊢ ( 𝑎 = ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) → ( 𝑎 ∩ 𝐴 ) = ( ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ∩ 𝐴 ) ) |
| 61 |
60
|
eqeq2d |
⊢ ( 𝑎 = ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) → ( 𝑐 = ( 𝑎 ∩ 𝐴 ) ↔ 𝑐 = ( ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ∩ 𝐴 ) ) ) |
| 62 |
59 61
|
anbi12d |
⊢ ( 𝑎 = ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) → ( ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ↔ ( ( ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ) ) ∧ 𝑐 = ( ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ∩ 𝐴 ) ) ) ) |
| 63 |
54 62
|
spcev |
⊢ ( ( ( ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ) ) ∧ 𝑐 = ( ( 𝑐 ∪ ( 𝑒 ∖ 𝐴 ) ) ∩ 𝐴 ) ) → ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) |
| 64 |
23 41 50 63
|
syl21anc |
⊢ ( ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ∧ 𝑒 ∈ 𝐽 ) ∧ 𝑑 = ( 𝑒 ∩ 𝐴 ) ) → ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) |
| 65 |
10
|
ad3antrrr |
⊢ ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) → 𝐽 ∈ Top ) |
| 66 |
10
|
uniexd |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → ∪ 𝐽 ∈ V ) |
| 67 |
1 66
|
eqeltrid |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → 𝑋 ∈ V ) |
| 68 |
67 11
|
ssexd |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → 𝐴 ∈ V ) |
| 69 |
68
|
ad3antrrr |
⊢ ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) → 𝐴 ∈ V ) |
| 70 |
|
simplr |
⊢ ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) → 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) |
| 71 |
|
elrest |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ V ) → ( 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ↔ ∃ 𝑒 ∈ 𝐽 𝑑 = ( 𝑒 ∩ 𝐴 ) ) ) |
| 72 |
71
|
biimpa |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ V ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) → ∃ 𝑒 ∈ 𝐽 𝑑 = ( 𝑒 ∩ 𝐴 ) ) |
| 73 |
65 69 70 72
|
syl21anc |
⊢ ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) → ∃ 𝑒 ∈ 𝐽 𝑑 = ( 𝑒 ∩ 𝐴 ) ) |
| 74 |
64 73
|
r19.29a |
⊢ ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) → ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) |
| 75 |
8 74
|
sylanl1 |
⊢ ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ ( 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ∧ ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ) ∧ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ) ∧ ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) → ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) |
| 76 |
|
simprr |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ ( 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ∧ ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ) → ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) |
| 77 |
6 75 76
|
r19.29af |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ ( 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ∧ ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ) → ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) |
| 78 |
|
inss2 |
⊢ ( 𝑎 ∩ 𝐴 ) ⊆ 𝐴 |
| 79 |
|
sseq1 |
⊢ ( 𝑐 = ( 𝑎 ∩ 𝐴 ) → ( 𝑐 ⊆ 𝐴 ↔ ( 𝑎 ∩ 𝐴 ) ⊆ 𝐴 ) ) |
| 80 |
78 79
|
mpbiri |
⊢ ( 𝑐 = ( 𝑎 ∩ 𝐴 ) → 𝑐 ⊆ 𝐴 ) |
| 81 |
80
|
adantl |
⊢ ( ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) → 𝑐 ⊆ 𝐴 ) |
| 82 |
81
|
exlimiv |
⊢ ( ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) → 𝑐 ⊆ 𝐴 ) |
| 83 |
82
|
adantl |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) → 𝑐 ⊆ 𝐴 ) |
| 84 |
13
|
adantr |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) → 𝐴 = ∪ ( 𝐽 ↾t 𝐴 ) ) |
| 85 |
83 84
|
sseqtrd |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) → 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) |
| 86 |
10
|
ad4antr |
⊢ ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ 𝐽 ) ∧ ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) → 𝐽 ∈ Top ) |
| 87 |
68
|
ad4antr |
⊢ ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ 𝐽 ) ∧ ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) → 𝐴 ∈ V ) |
| 88 |
|
simplr |
⊢ ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ 𝐽 ) ∧ ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) → 𝑏 ∈ 𝐽 ) |
| 89 |
|
elrestr |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ V ∧ 𝑏 ∈ 𝐽 ) → ( 𝑏 ∩ 𝐴 ) ∈ ( 𝐽 ↾t 𝐴 ) ) |
| 90 |
86 87 88 89
|
syl3anc |
⊢ ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ 𝐽 ) ∧ ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) → ( 𝑏 ∩ 𝐴 ) ∈ ( 𝐽 ↾t 𝐴 ) ) |
| 91 |
|
simprl |
⊢ ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ 𝐽 ) ∧ ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) → 𝐵 ⊆ 𝑏 ) |
| 92 |
|
simp3 |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ⊆ 𝐴 ) |
| 93 |
92
|
ad4antr |
⊢ ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ 𝐽 ) ∧ ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) → 𝐵 ⊆ 𝐴 ) |
| 94 |
91 93
|
ssind |
⊢ ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ 𝐽 ) ∧ ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) → 𝐵 ⊆ ( 𝑏 ∩ 𝐴 ) ) |
| 95 |
|
simprr |
⊢ ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ 𝐽 ) ∧ ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) → 𝑏 ⊆ 𝑎 ) |
| 96 |
95
|
ssrind |
⊢ ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ 𝐽 ) ∧ ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) → ( 𝑏 ∩ 𝐴 ) ⊆ ( 𝑎 ∩ 𝐴 ) ) |
| 97 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ 𝐽 ) ∧ ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) → 𝑐 = ( 𝑎 ∩ 𝐴 ) ) |
| 98 |
96 97
|
sseqtrrd |
⊢ ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ 𝐽 ) ∧ ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) → ( 𝑏 ∩ 𝐴 ) ⊆ 𝑐 ) |
| 99 |
90 94 98
|
jca32 |
⊢ ( ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ 𝐽 ) ∧ ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) → ( ( 𝑏 ∩ 𝐴 ) ∈ ( 𝐽 ↾t 𝐴 ) ∧ ( 𝐵 ⊆ ( 𝑏 ∩ 𝐴 ) ∧ ( 𝑏 ∩ 𝐴 ) ⊆ 𝑐 ) ) ) |
| 100 |
99
|
ex |
⊢ ( ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ∧ 𝑎 ⊆ 𝑋 ) ∧ 𝑏 ∈ 𝐽 ) → ( ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) → ( ( 𝑏 ∩ 𝐴 ) ∈ ( 𝐽 ↾t 𝐴 ) ∧ ( 𝐵 ⊆ ( 𝑏 ∩ 𝐴 ) ∧ ( 𝑏 ∩ 𝐴 ) ⊆ 𝑐 ) ) ) ) |
| 101 |
100
|
reximdva |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ∧ 𝑎 ⊆ 𝑋 ) → ( ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) → ∃ 𝑏 ∈ 𝐽 ( ( 𝑏 ∩ 𝐴 ) ∈ ( 𝐽 ↾t 𝐴 ) ∧ ( 𝐵 ⊆ ( 𝑏 ∩ 𝐴 ) ∧ ( 𝑏 ∩ 𝐴 ) ⊆ 𝑐 ) ) ) ) |
| 102 |
101
|
impr |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ∧ ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ) → ∃ 𝑏 ∈ 𝐽 ( ( 𝑏 ∩ 𝐴 ) ∈ ( 𝐽 ↾t 𝐴 ) ∧ ( 𝐵 ⊆ ( 𝑏 ∩ 𝐴 ) ∧ ( 𝑏 ∩ 𝐴 ) ⊆ 𝑐 ) ) ) |
| 103 |
102
|
an32s |
⊢ ( ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) → ∃ 𝑏 ∈ 𝐽 ( ( 𝑏 ∩ 𝐴 ) ∈ ( 𝐽 ↾t 𝐴 ) ∧ ( 𝐵 ⊆ ( 𝑏 ∩ 𝐴 ) ∧ ( 𝑏 ∩ 𝐴 ) ⊆ 𝑐 ) ) ) |
| 104 |
103
|
expl |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → ( ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) → ∃ 𝑏 ∈ 𝐽 ( ( 𝑏 ∩ 𝐴 ) ∈ ( 𝐽 ↾t 𝐴 ) ∧ ( 𝐵 ⊆ ( 𝑏 ∩ 𝐴 ) ∧ ( 𝑏 ∩ 𝐴 ) ⊆ 𝑐 ) ) ) ) |
| 105 |
104
|
exlimdv |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → ( ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) → ∃ 𝑏 ∈ 𝐽 ( ( 𝑏 ∩ 𝐴 ) ∈ ( 𝐽 ↾t 𝐴 ) ∧ ( 𝐵 ⊆ ( 𝑏 ∩ 𝐴 ) ∧ ( 𝑏 ∩ 𝐴 ) ⊆ 𝑐 ) ) ) ) |
| 106 |
105
|
imp |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) → ∃ 𝑏 ∈ 𝐽 ( ( 𝑏 ∩ 𝐴 ) ∈ ( 𝐽 ↾t 𝐴 ) ∧ ( 𝐵 ⊆ ( 𝑏 ∩ 𝐴 ) ∧ ( 𝑏 ∩ 𝐴 ) ⊆ 𝑐 ) ) ) |
| 107 |
|
sseq2 |
⊢ ( 𝑑 = ( 𝑏 ∩ 𝐴 ) → ( 𝐵 ⊆ 𝑑 ↔ 𝐵 ⊆ ( 𝑏 ∩ 𝐴 ) ) ) |
| 108 |
|
sseq1 |
⊢ ( 𝑑 = ( 𝑏 ∩ 𝐴 ) → ( 𝑑 ⊆ 𝑐 ↔ ( 𝑏 ∩ 𝐴 ) ⊆ 𝑐 ) ) |
| 109 |
107 108
|
anbi12d |
⊢ ( 𝑑 = ( 𝑏 ∩ 𝐴 ) → ( ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ↔ ( 𝐵 ⊆ ( 𝑏 ∩ 𝐴 ) ∧ ( 𝑏 ∩ 𝐴 ) ⊆ 𝑐 ) ) ) |
| 110 |
109
|
rspcev |
⊢ ( ( ( 𝑏 ∩ 𝐴 ) ∈ ( 𝐽 ↾t 𝐴 ) ∧ ( 𝐵 ⊆ ( 𝑏 ∩ 𝐴 ) ∧ ( 𝑏 ∩ 𝐴 ) ⊆ 𝑐 ) ) → ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) |
| 111 |
110
|
rexlimivw |
⊢ ( ∃ 𝑏 ∈ 𝐽 ( ( 𝑏 ∩ 𝐴 ) ∈ ( 𝐽 ↾t 𝐴 ) ∧ ( 𝐵 ⊆ ( 𝑏 ∩ 𝐴 ) ∧ ( 𝑏 ∩ 𝐴 ) ⊆ 𝑐 ) ) → ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) |
| 112 |
106 111
|
syl |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) → ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) |
| 113 |
85 112
|
jca |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) ∧ ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) → ( 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ∧ ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ) |
| 114 |
77 113
|
impbida |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → ( ( 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ∧ ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ↔ ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) ) |
| 115 |
|
resttop |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ∈ V ) → ( 𝐽 ↾t 𝐴 ) ∈ Top ) |
| 116 |
10 68 115
|
syl2anc |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → ( 𝐽 ↾t 𝐴 ) ∈ Top ) |
| 117 |
92 13
|
sseqtrd |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) |
| 118 |
|
eqid |
⊢ ∪ ( 𝐽 ↾t 𝐴 ) = ∪ ( 𝐽 ↾t 𝐴 ) |
| 119 |
118
|
isnei |
⊢ ( ( ( 𝐽 ↾t 𝐴 ) ∈ Top ∧ 𝐵 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ) → ( 𝑐 ∈ ( ( nei ‘ ( 𝐽 ↾t 𝐴 ) ) ‘ 𝐵 ) ↔ ( 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ∧ ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ) ) |
| 120 |
116 117 119
|
syl2anc |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → ( 𝑐 ∈ ( ( nei ‘ ( 𝐽 ↾t 𝐴 ) ) ‘ 𝐵 ) ↔ ( 𝑐 ⊆ ∪ ( 𝐽 ↾t 𝐴 ) ∧ ∃ 𝑑 ∈ ( 𝐽 ↾t 𝐴 ) ( 𝐵 ⊆ 𝑑 ∧ 𝑑 ⊆ 𝑐 ) ) ) ) |
| 121 |
|
fvex |
⊢ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ∈ V |
| 122 |
|
restval |
⊢ ( ( ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ∈ V ∧ 𝐴 ∈ V ) → ( ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↾t 𝐴 ) = ran ( 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↦ ( 𝑎 ∩ 𝐴 ) ) ) |
| 123 |
121 68 122
|
sylancr |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → ( ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↾t 𝐴 ) = ran ( 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↦ ( 𝑎 ∩ 𝐴 ) ) ) |
| 124 |
123
|
eleq2d |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → ( 𝑐 ∈ ( ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↾t 𝐴 ) ↔ 𝑐 ∈ ran ( 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↦ ( 𝑎 ∩ 𝐴 ) ) ) ) |
| 125 |
92 11
|
sstrd |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ⊆ 𝑋 ) |
| 126 |
|
eqid |
⊢ ( 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↦ ( 𝑎 ∩ 𝐴 ) ) = ( 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↦ ( 𝑎 ∩ 𝐴 ) ) |
| 127 |
126
|
elrnmpt |
⊢ ( 𝑐 ∈ V → ( 𝑐 ∈ ran ( 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↦ ( 𝑎 ∩ 𝐴 ) ) ↔ ∃ 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) |
| 128 |
127
|
elv |
⊢ ( 𝑐 ∈ ran ( 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↦ ( 𝑎 ∩ 𝐴 ) ) ↔ ∃ 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) 𝑐 = ( 𝑎 ∩ 𝐴 ) ) |
| 129 |
|
df-rex |
⊢ ( ∃ 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) 𝑐 = ( 𝑎 ∩ 𝐴 ) ↔ ∃ 𝑎 ( 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) |
| 130 |
128 129
|
bitri |
⊢ ( 𝑐 ∈ ran ( 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↦ ( 𝑎 ∩ 𝐴 ) ) ↔ ∃ 𝑎 ( 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) |
| 131 |
1
|
isnei |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋 ) → ( 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↔ ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ) ) |
| 132 |
131
|
anbi1d |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋 ) → ( ( 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ↔ ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) ) |
| 133 |
132
|
exbidv |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋 ) → ( ∃ 𝑎 ( 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ↔ ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) ) |
| 134 |
130 133
|
syl5bb |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐵 ⊆ 𝑋 ) → ( 𝑐 ∈ ran ( 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↦ ( 𝑎 ∩ 𝐴 ) ) ↔ ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) ) |
| 135 |
10 125 134
|
syl2anc |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → ( 𝑐 ∈ ran ( 𝑎 ∈ ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↦ ( 𝑎 ∩ 𝐴 ) ) ↔ ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) ) |
| 136 |
124 135
|
bitrd |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → ( 𝑐 ∈ ( ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↾t 𝐴 ) ↔ ∃ 𝑎 ( ( 𝑎 ⊆ 𝑋 ∧ ∃ 𝑏 ∈ 𝐽 ( 𝐵 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑎 ) ) ∧ 𝑐 = ( 𝑎 ∩ 𝐴 ) ) ) ) |
| 137 |
114 120 136
|
3bitr4d |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → ( 𝑐 ∈ ( ( nei ‘ ( 𝐽 ↾t 𝐴 ) ) ‘ 𝐵 ) ↔ 𝑐 ∈ ( ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↾t 𝐴 ) ) ) |
| 138 |
137
|
eqrdv |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴 ) → ( ( nei ‘ ( 𝐽 ↾t 𝐴 ) ) ‘ 𝐵 ) = ( ( ( nei ‘ 𝐽 ) ‘ 𝐵 ) ↾t 𝐴 ) ) |