Step |
Hyp |
Ref |
Expression |
1 |
|
alexsub.1 |
⊢ ( 𝜑 → 𝑋 ∈ UFL ) |
2 |
|
alexsub.2 |
⊢ ( 𝜑 → 𝑋 = ∪ 𝐵 ) |
3 |
|
alexsub.3 |
⊢ ( 𝜑 → 𝐽 = ( topGen ‘ ( fi ‘ 𝐵 ) ) ) |
4 |
|
alexsub.4 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥 ) ) → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) |
5 |
|
alexsub.5 |
⊢ ( 𝜑 → 𝐹 ∈ ( UFil ‘ 𝑋 ) ) |
6 |
|
alexsub.6 |
⊢ ( 𝜑 → ( 𝐽 fLim 𝐹 ) = ∅ ) |
7 |
|
eldif |
⊢ ( 𝑥 ∈ ( 𝑋 ∖ ∪ ( 𝐵 ∖ 𝐹 ) ) ↔ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) |
8 |
3
|
eleq2d |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐽 ↔ 𝑦 ∈ ( topGen ‘ ( fi ‘ 𝐵 ) ) ) ) |
9 |
8
|
anbi1d |
⊢ ( 𝜑 → ( ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ↔ ( 𝑦 ∈ ( topGen ‘ ( fi ‘ 𝐵 ) ) ∧ 𝑥 ∈ 𝑦 ) ) ) |
10 |
9
|
biimpa |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) → ( 𝑦 ∈ ( topGen ‘ ( fi ‘ 𝐵 ) ) ∧ 𝑥 ∈ 𝑦 ) ) |
11 |
10
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) → ( 𝑦 ∈ ( topGen ‘ ( fi ‘ 𝐵 ) ) ∧ 𝑥 ∈ 𝑦 ) ) |
12 |
|
tg2 |
⊢ ( ( 𝑦 ∈ ( topGen ‘ ( fi ‘ 𝐵 ) ) ∧ 𝑥 ∈ 𝑦 ) → ∃ 𝑧 ∈ ( fi ‘ 𝐵 ) ( 𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) |
13 |
11 12
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) → ∃ 𝑧 ∈ ( fi ‘ 𝐵 ) ( 𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) |
14 |
|
ufilfil |
⊢ ( 𝐹 ∈ ( UFil ‘ 𝑋 ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) ) |
15 |
5 14
|
syl |
⊢ ( 𝜑 → 𝐹 ∈ ( Fil ‘ 𝑋 ) ) |
16 |
15
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) ∧ ( 𝑧 ∈ ( fi ‘ 𝐵 ) ∧ ( 𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) ) |
17 |
5
|
elfvexd |
⊢ ( 𝜑 → 𝑋 ∈ V ) |
18 |
2 17
|
eqeltrrd |
⊢ ( 𝜑 → ∪ 𝐵 ∈ V ) |
19 |
|
uniexb |
⊢ ( 𝐵 ∈ V ↔ ∪ 𝐵 ∈ V ) |
20 |
18 19
|
sylibr |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
21 |
|
elfi2 |
⊢ ( 𝐵 ∈ V → ( 𝑧 ∈ ( fi ‘ 𝐵 ) ↔ ∃ 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) 𝑧 = ∩ 𝑦 ) ) |
22 |
20 21
|
syl |
⊢ ( 𝜑 → ( 𝑧 ∈ ( fi ‘ 𝐵 ) ↔ ∃ 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) 𝑧 = ∩ 𝑦 ) ) |
23 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) → ( 𝑧 ∈ ( fi ‘ 𝐵 ) ↔ ∃ 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) 𝑧 = ∩ 𝑦 ) ) |
24 |
15
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) ) |
25 |
|
simplrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ∧ 𝑧 ∈ 𝑦 ) ) → ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) |
26 |
|
intss1 |
⊢ ( 𝑧 ∈ 𝑦 → ∩ 𝑦 ⊆ 𝑧 ) |
27 |
26
|
adantl |
⊢ ( ( ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ∧ 𝑧 ∈ 𝑦 ) → ∩ 𝑦 ⊆ 𝑧 ) |
28 |
|
simplr |
⊢ ( ( ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ∧ 𝑧 ∈ 𝑦 ) → 𝑥 ∈ ∩ 𝑦 ) |
29 |
27 28
|
sseldd |
⊢ ( ( ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ∧ 𝑧 ∈ 𝑦 ) → 𝑥 ∈ 𝑧 ) |
30 |
29
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ∧ 𝑧 ∈ 𝑦 ) ) ∧ ¬ 𝑧 ∈ 𝐹 ) → 𝑥 ∈ 𝑧 ) |
31 |
|
eldifsn |
⊢ ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ↔ ( 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) ∧ 𝑦 ≠ ∅ ) ) |
32 |
31
|
simplbi |
⊢ ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) → 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) ) |
33 |
32
|
ad2antrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ) → 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) ) |
34 |
|
elfpw |
⊢ ( 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) ↔ ( 𝑦 ⊆ 𝐵 ∧ 𝑦 ∈ Fin ) ) |
35 |
34
|
simplbi |
⊢ ( 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) → 𝑦 ⊆ 𝐵 ) |
36 |
33 35
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ) → 𝑦 ⊆ 𝐵 ) |
37 |
36
|
sselda |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ) ∧ 𝑧 ∈ 𝑦 ) → 𝑧 ∈ 𝐵 ) |
38 |
37
|
anasss |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ∧ 𝑧 ∈ 𝑦 ) ) → 𝑧 ∈ 𝐵 ) |
39 |
38
|
anim1i |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ∧ 𝑧 ∈ 𝑦 ) ) ∧ ¬ 𝑧 ∈ 𝐹 ) → ( 𝑧 ∈ 𝐵 ∧ ¬ 𝑧 ∈ 𝐹 ) ) |
40 |
|
eldif |
⊢ ( 𝑧 ∈ ( 𝐵 ∖ 𝐹 ) ↔ ( 𝑧 ∈ 𝐵 ∧ ¬ 𝑧 ∈ 𝐹 ) ) |
41 |
39 40
|
sylibr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ∧ 𝑧 ∈ 𝑦 ) ) ∧ ¬ 𝑧 ∈ 𝐹 ) → 𝑧 ∈ ( 𝐵 ∖ 𝐹 ) ) |
42 |
|
elunii |
⊢ ( ( 𝑥 ∈ 𝑧 ∧ 𝑧 ∈ ( 𝐵 ∖ 𝐹 ) ) → 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) |
43 |
30 41 42
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ∧ 𝑧 ∈ 𝑦 ) ) ∧ ¬ 𝑧 ∈ 𝐹 ) → 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) |
44 |
43
|
ex |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ∧ 𝑧 ∈ 𝑦 ) ) → ( ¬ 𝑧 ∈ 𝐹 → 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) |
45 |
25 44
|
mt3d |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ∧ 𝑧 ∈ 𝑦 ) ) → 𝑧 ∈ 𝐹 ) |
46 |
45
|
expr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ) → ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐹 ) ) |
47 |
46
|
ssrdv |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ) → 𝑦 ⊆ 𝐹 ) |
48 |
|
eldifsni |
⊢ ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) → 𝑦 ≠ ∅ ) |
49 |
48
|
ad2antrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ) → 𝑦 ≠ ∅ ) |
50 |
|
elinel2 |
⊢ ( 𝑦 ∈ ( 𝒫 𝐵 ∩ Fin ) → 𝑦 ∈ Fin ) |
51 |
33 50
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ) → 𝑦 ∈ Fin ) |
52 |
|
elfir |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ( 𝑦 ⊆ 𝐹 ∧ 𝑦 ≠ ∅ ∧ 𝑦 ∈ Fin ) ) → ∩ 𝑦 ∈ ( fi ‘ 𝐹 ) ) |
53 |
24 47 49 51 52
|
syl13anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ) → ∩ 𝑦 ∈ ( fi ‘ 𝐹 ) ) |
54 |
|
filfi |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → ( fi ‘ 𝐹 ) = 𝐹 ) |
55 |
24 54
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ) → ( fi ‘ 𝐹 ) = 𝐹 ) |
56 |
53 55
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ∧ 𝑥 ∈ ∩ 𝑦 ) ) → ∩ 𝑦 ∈ 𝐹 ) |
57 |
56
|
expr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ) → ( 𝑥 ∈ ∩ 𝑦 → ∩ 𝑦 ∈ 𝐹 ) ) |
58 |
|
eleq2 |
⊢ ( 𝑧 = ∩ 𝑦 → ( 𝑥 ∈ 𝑧 ↔ 𝑥 ∈ ∩ 𝑦 ) ) |
59 |
|
eleq1 |
⊢ ( 𝑧 = ∩ 𝑦 → ( 𝑧 ∈ 𝐹 ↔ ∩ 𝑦 ∈ 𝐹 ) ) |
60 |
58 59
|
imbi12d |
⊢ ( 𝑧 = ∩ 𝑦 → ( ( 𝑥 ∈ 𝑧 → 𝑧 ∈ 𝐹 ) ↔ ( 𝑥 ∈ ∩ 𝑦 → ∩ 𝑦 ∈ 𝐹 ) ) ) |
61 |
57 60
|
syl5ibrcom |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) ) → ( 𝑧 = ∩ 𝑦 → ( 𝑥 ∈ 𝑧 → 𝑧 ∈ 𝐹 ) ) ) |
62 |
61
|
rexlimdva |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) → ( ∃ 𝑦 ∈ ( ( 𝒫 𝐵 ∩ Fin ) ∖ { ∅ } ) 𝑧 = ∩ 𝑦 → ( 𝑥 ∈ 𝑧 → 𝑧 ∈ 𝐹 ) ) ) |
63 |
23 62
|
sylbid |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) → ( 𝑧 ∈ ( fi ‘ 𝐵 ) → ( 𝑥 ∈ 𝑧 → 𝑧 ∈ 𝐹 ) ) ) |
64 |
63
|
imp32 |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑧 ∈ ( fi ‘ 𝐵 ) ∧ 𝑥 ∈ 𝑧 ) ) → 𝑧 ∈ 𝐹 ) |
65 |
64
|
adantrrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑧 ∈ ( fi ‘ 𝐵 ) ∧ ( 𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) → 𝑧 ∈ 𝐹 ) |
66 |
65
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) ∧ ( 𝑧 ∈ ( fi ‘ 𝐵 ) ∧ ( 𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) → 𝑧 ∈ 𝐹 ) |
67 |
|
elssuni |
⊢ ( 𝑦 ∈ 𝐽 → 𝑦 ⊆ ∪ 𝐽 ) |
68 |
67
|
ad2antrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) → 𝑦 ⊆ ∪ 𝐽 ) |
69 |
|
fibas |
⊢ ( fi ‘ 𝐵 ) ∈ TopBases |
70 |
|
tgtopon |
⊢ ( ( fi ‘ 𝐵 ) ∈ TopBases → ( topGen ‘ ( fi ‘ 𝐵 ) ) ∈ ( TopOn ‘ ∪ ( fi ‘ 𝐵 ) ) ) |
71 |
69 70
|
ax-mp |
⊢ ( topGen ‘ ( fi ‘ 𝐵 ) ) ∈ ( TopOn ‘ ∪ ( fi ‘ 𝐵 ) ) |
72 |
3 71
|
eqeltrdi |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ ∪ ( fi ‘ 𝐵 ) ) ) |
73 |
|
fiuni |
⊢ ( 𝐵 ∈ V → ∪ 𝐵 = ∪ ( fi ‘ 𝐵 ) ) |
74 |
20 73
|
syl |
⊢ ( 𝜑 → ∪ 𝐵 = ∪ ( fi ‘ 𝐵 ) ) |
75 |
2 74
|
eqtrd |
⊢ ( 𝜑 → 𝑋 = ∪ ( fi ‘ 𝐵 ) ) |
76 |
75
|
fveq2d |
⊢ ( 𝜑 → ( TopOn ‘ 𝑋 ) = ( TopOn ‘ ∪ ( fi ‘ 𝐵 ) ) ) |
77 |
72 76
|
eleqtrrd |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
78 |
|
toponuni |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 ) |
79 |
77 78
|
syl |
⊢ ( 𝜑 → 𝑋 = ∪ 𝐽 ) |
80 |
79
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) → 𝑋 = ∪ 𝐽 ) |
81 |
68 80
|
sseqtrrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) → 𝑦 ⊆ 𝑋 ) |
82 |
81
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) ∧ ( 𝑧 ∈ ( fi ‘ 𝐵 ) ∧ ( 𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) → 𝑦 ⊆ 𝑋 ) |
83 |
|
simprrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) ∧ ( 𝑧 ∈ ( fi ‘ 𝐵 ) ∧ ( 𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) → 𝑧 ⊆ 𝑦 ) |
84 |
|
filss |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ( 𝑧 ∈ 𝐹 ∧ 𝑦 ⊆ 𝑋 ∧ 𝑧 ⊆ 𝑦 ) ) → 𝑦 ∈ 𝐹 ) |
85 |
16 66 82 83 84
|
syl13anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) ∧ ( 𝑧 ∈ ( fi ‘ 𝐵 ) ∧ ( 𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) → 𝑦 ∈ 𝐹 ) |
86 |
13 85
|
rexlimddv |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ ( 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) ) → 𝑦 ∈ 𝐹 ) |
87 |
86
|
expr |
⊢ ( ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) ∧ 𝑦 ∈ 𝐽 ) → ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹 ) ) |
88 |
87
|
ralrimiva |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) ) → ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹 ) ) |
89 |
88
|
expr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) → ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹 ) ) ) |
90 |
89
|
imdistanda |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ ∪ ( 𝐵 ∖ 𝐹 ) ) → ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹 ) ) ) ) |
91 |
7 90
|
syl5bi |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝑋 ∖ ∪ ( 𝐵 ∖ 𝐹 ) ) → ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹 ) ) ) ) |
92 |
|
flimopn |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝑥 ∈ ( 𝐽 fLim 𝐹 ) ↔ ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹 ) ) ) ) |
93 |
77 15 92
|
syl2anc |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐽 fLim 𝐹 ) ↔ ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐹 ) ) ) ) |
94 |
91 93
|
sylibrd |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝑋 ∖ ∪ ( 𝐵 ∖ 𝐹 ) ) → 𝑥 ∈ ( 𝐽 fLim 𝐹 ) ) ) |
95 |
94
|
ssrdv |
⊢ ( 𝜑 → ( 𝑋 ∖ ∪ ( 𝐵 ∖ 𝐹 ) ) ⊆ ( 𝐽 fLim 𝐹 ) ) |
96 |
|
sseq0 |
⊢ ( ( ( 𝑋 ∖ ∪ ( 𝐵 ∖ 𝐹 ) ) ⊆ ( 𝐽 fLim 𝐹 ) ∧ ( 𝐽 fLim 𝐹 ) = ∅ ) → ( 𝑋 ∖ ∪ ( 𝐵 ∖ 𝐹 ) ) = ∅ ) |
97 |
95 6 96
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 ∖ ∪ ( 𝐵 ∖ 𝐹 ) ) = ∅ ) |
98 |
|
ssdif0 |
⊢ ( 𝑋 ⊆ ∪ ( 𝐵 ∖ 𝐹 ) ↔ ( 𝑋 ∖ ∪ ( 𝐵 ∖ 𝐹 ) ) = ∅ ) |
99 |
97 98
|
sylibr |
⊢ ( 𝜑 → 𝑋 ⊆ ∪ ( 𝐵 ∖ 𝐹 ) ) |
100 |
|
difss |
⊢ ( 𝐵 ∖ 𝐹 ) ⊆ 𝐵 |
101 |
100
|
unissi |
⊢ ∪ ( 𝐵 ∖ 𝐹 ) ⊆ ∪ 𝐵 |
102 |
101 2
|
sseqtrrid |
⊢ ( 𝜑 → ∪ ( 𝐵 ∖ 𝐹 ) ⊆ 𝑋 ) |
103 |
99 102
|
eqssd |
⊢ ( 𝜑 → 𝑋 = ∪ ( 𝐵 ∖ 𝐹 ) ) |
104 |
103 100
|
jctil |
⊢ ( 𝜑 → ( ( 𝐵 ∖ 𝐹 ) ⊆ 𝐵 ∧ 𝑋 = ∪ ( 𝐵 ∖ 𝐹 ) ) ) |
105 |
20
|
difexd |
⊢ ( 𝜑 → ( 𝐵 ∖ 𝐹 ) ∈ V ) |
106 |
105
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝐵 ∖ 𝐹 ) ⊆ 𝐵 ∧ 𝑋 = ∪ ( 𝐵 ∖ 𝐹 ) ) ) → ( 𝐵 ∖ 𝐹 ) ∈ V ) |
107 |
|
sseq1 |
⊢ ( 𝑥 = ( 𝐵 ∖ 𝐹 ) → ( 𝑥 ⊆ 𝐵 ↔ ( 𝐵 ∖ 𝐹 ) ⊆ 𝐵 ) ) |
108 |
|
unieq |
⊢ ( 𝑥 = ( 𝐵 ∖ 𝐹 ) → ∪ 𝑥 = ∪ ( 𝐵 ∖ 𝐹 ) ) |
109 |
108
|
eqeq2d |
⊢ ( 𝑥 = ( 𝐵 ∖ 𝐹 ) → ( 𝑋 = ∪ 𝑥 ↔ 𝑋 = ∪ ( 𝐵 ∖ 𝐹 ) ) ) |
110 |
107 109
|
anbi12d |
⊢ ( 𝑥 = ( 𝐵 ∖ 𝐹 ) → ( ( 𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥 ) ↔ ( ( 𝐵 ∖ 𝐹 ) ⊆ 𝐵 ∧ 𝑋 = ∪ ( 𝐵 ∖ 𝐹 ) ) ) ) |
111 |
110
|
anbi2d |
⊢ ( 𝑥 = ( 𝐵 ∖ 𝐹 ) → ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥 ) ) ↔ ( 𝜑 ∧ ( ( 𝐵 ∖ 𝐹 ) ⊆ 𝐵 ∧ 𝑋 = ∪ ( 𝐵 ∖ 𝐹 ) ) ) ) ) |
112 |
|
pweq |
⊢ ( 𝑥 = ( 𝐵 ∖ 𝐹 ) → 𝒫 𝑥 = 𝒫 ( 𝐵 ∖ 𝐹 ) ) |
113 |
112
|
ineq1d |
⊢ ( 𝑥 = ( 𝐵 ∖ 𝐹 ) → ( 𝒫 𝑥 ∩ Fin ) = ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) |
114 |
113
|
rexeqdv |
⊢ ( 𝑥 = ( 𝐵 ∖ 𝐹 ) → ( ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ↔ ∃ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) |
115 |
111 114
|
imbi12d |
⊢ ( 𝑥 = ( 𝐵 ∖ 𝐹 ) → ( ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥 ) ) → ∃ 𝑦 ∈ ( 𝒫 𝑥 ∩ Fin ) 𝑋 = ∪ 𝑦 ) ↔ ( ( 𝜑 ∧ ( ( 𝐵 ∖ 𝐹 ) ⊆ 𝐵 ∧ 𝑋 = ∪ ( 𝐵 ∖ 𝐹 ) ) ) → ∃ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) ) |
116 |
115 4
|
vtoclg |
⊢ ( ( 𝐵 ∖ 𝐹 ) ∈ V → ( ( 𝜑 ∧ ( ( 𝐵 ∖ 𝐹 ) ⊆ 𝐵 ∧ 𝑋 = ∪ ( 𝐵 ∖ 𝐹 ) ) ) → ∃ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) 𝑋 = ∪ 𝑦 ) ) |
117 |
106 116
|
mpcom |
⊢ ( ( 𝜑 ∧ ( ( 𝐵 ∖ 𝐹 ) ⊆ 𝐵 ∧ 𝑋 = ∪ ( 𝐵 ∖ 𝐹 ) ) ) → ∃ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) 𝑋 = ∪ 𝑦 ) |
118 |
104 117
|
mpdan |
⊢ ( 𝜑 → ∃ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) 𝑋 = ∪ 𝑦 ) |
119 |
|
unieq |
⊢ ( 𝑦 = ∅ → ∪ 𝑦 = ∪ ∅ ) |
120 |
|
uni0 |
⊢ ∪ ∅ = ∅ |
121 |
119 120
|
eqtrdi |
⊢ ( 𝑦 = ∅ → ∪ 𝑦 = ∅ ) |
122 |
121
|
neeq2d |
⊢ ( 𝑦 = ∅ → ( 𝑋 ≠ ∪ 𝑦 ↔ 𝑋 ≠ ∅ ) ) |
123 |
|
difssd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) → ( 𝑋 ∖ 𝑧 ) ⊆ 𝑋 ) |
124 |
123
|
ralrimivw |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) → ∀ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ⊆ 𝑋 ) |
125 |
|
riinn0 |
⊢ ( ( ∀ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ⊆ 𝑋 ∧ 𝑦 ≠ ∅ ) → ( 𝑋 ∩ ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ) = ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ) |
126 |
124 125
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → ( 𝑋 ∩ ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ) = ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ) |
127 |
17
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → 𝑋 ∈ V ) |
128 |
127
|
difexd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → ( 𝑋 ∖ 𝑧 ) ∈ V ) |
129 |
128
|
ralrimivw |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → ∀ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ∈ V ) |
130 |
|
dfiin2g |
⊢ ( ∀ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ∈ V → ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) = ∩ { 𝑥 ∣ ∃ 𝑧 ∈ 𝑦 𝑥 = ( 𝑋 ∖ 𝑧 ) } ) |
131 |
129 130
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) = ∩ { 𝑥 ∣ ∃ 𝑧 ∈ 𝑦 𝑥 = ( 𝑋 ∖ 𝑧 ) } ) |
132 |
|
eqid |
⊢ ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) = ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) |
133 |
132
|
rnmpt |
⊢ ran ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) = { 𝑥 ∣ ∃ 𝑧 ∈ 𝑦 𝑥 = ( 𝑋 ∖ 𝑧 ) } |
134 |
133
|
inteqi |
⊢ ∩ ran ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) = ∩ { 𝑥 ∣ ∃ 𝑧 ∈ 𝑦 𝑥 = ( 𝑋 ∖ 𝑧 ) } |
135 |
131 134
|
eqtr4di |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) = ∩ ran ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) ) |
136 |
126 135
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → ( 𝑋 ∩ ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ) = ∩ ran ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) ) |
137 |
15
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) ) |
138 |
|
elfpw |
⊢ ( 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ↔ ( 𝑦 ⊆ ( 𝐵 ∖ 𝐹 ) ∧ 𝑦 ∈ Fin ) ) |
139 |
138
|
simplbi |
⊢ ( 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) → 𝑦 ⊆ ( 𝐵 ∖ 𝐹 ) ) |
140 |
139
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → 𝑦 ⊆ ( 𝐵 ∖ 𝐹 ) ) |
141 |
140
|
sselda |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) ∧ 𝑧 ∈ 𝑦 ) → 𝑧 ∈ ( 𝐵 ∖ 𝐹 ) ) |
142 |
141
|
eldifbd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) ∧ 𝑧 ∈ 𝑦 ) → ¬ 𝑧 ∈ 𝐹 ) |
143 |
5
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) ∧ 𝑧 ∈ 𝑦 ) → 𝐹 ∈ ( UFil ‘ 𝑋 ) ) |
144 |
140
|
difss2d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → 𝑦 ⊆ 𝐵 ) |
145 |
144
|
sselda |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) ∧ 𝑧 ∈ 𝑦 ) → 𝑧 ∈ 𝐵 ) |
146 |
|
elssuni |
⊢ ( 𝑧 ∈ 𝐵 → 𝑧 ⊆ ∪ 𝐵 ) |
147 |
145 146
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) ∧ 𝑧 ∈ 𝑦 ) → 𝑧 ⊆ ∪ 𝐵 ) |
148 |
2
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) ∧ 𝑧 ∈ 𝑦 ) → 𝑋 = ∪ 𝐵 ) |
149 |
147 148
|
sseqtrrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) ∧ 𝑧 ∈ 𝑦 ) → 𝑧 ⊆ 𝑋 ) |
150 |
|
ufilb |
⊢ ( ( 𝐹 ∈ ( UFil ‘ 𝑋 ) ∧ 𝑧 ⊆ 𝑋 ) → ( ¬ 𝑧 ∈ 𝐹 ↔ ( 𝑋 ∖ 𝑧 ) ∈ 𝐹 ) ) |
151 |
143 149 150
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) ∧ 𝑧 ∈ 𝑦 ) → ( ¬ 𝑧 ∈ 𝐹 ↔ ( 𝑋 ∖ 𝑧 ) ∈ 𝐹 ) ) |
152 |
142 151
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) ∧ 𝑧 ∈ 𝑦 ) → ( 𝑋 ∖ 𝑧 ) ∈ 𝐹 ) |
153 |
152
|
fmpttd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) : 𝑦 ⟶ 𝐹 ) |
154 |
153
|
frnd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → ran ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) ⊆ 𝐹 ) |
155 |
132 152
|
dmmptd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → dom ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) = 𝑦 ) |
156 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → 𝑦 ≠ ∅ ) |
157 |
155 156
|
eqnetrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → dom ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) ≠ ∅ ) |
158 |
|
dm0rn0 |
⊢ ( dom ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) = ∅ ↔ ran ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) = ∅ ) |
159 |
158
|
necon3bii |
⊢ ( dom ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) ≠ ∅ ↔ ran ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) ≠ ∅ ) |
160 |
157 159
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → ran ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) ≠ ∅ ) |
161 |
|
elinel2 |
⊢ ( 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) → 𝑦 ∈ Fin ) |
162 |
161
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → 𝑦 ∈ Fin ) |
163 |
|
abrexfi |
⊢ ( 𝑦 ∈ Fin → { 𝑥 ∣ ∃ 𝑧 ∈ 𝑦 𝑥 = ( 𝑋 ∖ 𝑧 ) } ∈ Fin ) |
164 |
133 163
|
eqeltrid |
⊢ ( 𝑦 ∈ Fin → ran ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) ∈ Fin ) |
165 |
162 164
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → ran ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) ∈ Fin ) |
166 |
|
filintn0 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ( ran ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) ⊆ 𝐹 ∧ ran ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) ≠ ∅ ∧ ran ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) ∈ Fin ) ) → ∩ ran ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) ≠ ∅ ) |
167 |
137 154 160 165 166
|
syl13anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → ∩ ran ( 𝑧 ∈ 𝑦 ↦ ( 𝑋 ∖ 𝑧 ) ) ≠ ∅ ) |
168 |
136 167
|
eqnetrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → ( 𝑋 ∩ ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ) ≠ ∅ ) |
169 |
|
disj3 |
⊢ ( ( 𝑋 ∩ ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ) = ∅ ↔ 𝑋 = ( 𝑋 ∖ ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ) ) |
170 |
169
|
necon3bii |
⊢ ( ( 𝑋 ∩ ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ) ≠ ∅ ↔ 𝑋 ≠ ( 𝑋 ∖ ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ) ) |
171 |
168 170
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → 𝑋 ≠ ( 𝑋 ∖ ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ) ) |
172 |
|
iundif2 |
⊢ ∪ 𝑧 ∈ 𝑦 ( 𝑋 ∖ ( 𝑋 ∖ 𝑧 ) ) = ( 𝑋 ∖ ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ) |
173 |
|
dfss4 |
⊢ ( 𝑧 ⊆ 𝑋 ↔ ( 𝑋 ∖ ( 𝑋 ∖ 𝑧 ) ) = 𝑧 ) |
174 |
149 173
|
sylib |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) ∧ 𝑧 ∈ 𝑦 ) → ( 𝑋 ∖ ( 𝑋 ∖ 𝑧 ) ) = 𝑧 ) |
175 |
174
|
iuneq2dv |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → ∪ 𝑧 ∈ 𝑦 ( 𝑋 ∖ ( 𝑋 ∖ 𝑧 ) ) = ∪ 𝑧 ∈ 𝑦 𝑧 ) |
176 |
|
uniiun |
⊢ ∪ 𝑦 = ∪ 𝑧 ∈ 𝑦 𝑧 |
177 |
175 176
|
eqtr4di |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → ∪ 𝑧 ∈ 𝑦 ( 𝑋 ∖ ( 𝑋 ∖ 𝑧 ) ) = ∪ 𝑦 ) |
178 |
172 177
|
eqtr3id |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → ( 𝑋 ∖ ∩ 𝑧 ∈ 𝑦 ( 𝑋 ∖ 𝑧 ) ) = ∪ 𝑦 ) |
179 |
171 178
|
neeqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) ∧ 𝑦 ≠ ∅ ) → 𝑋 ≠ ∪ 𝑦 ) |
180 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) → 𝐹 ∈ ( Fil ‘ 𝑋 ) ) |
181 |
|
filtop |
⊢ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) → 𝑋 ∈ 𝐹 ) |
182 |
|
fileln0 |
⊢ ( ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑋 ∈ 𝐹 ) → 𝑋 ≠ ∅ ) |
183 |
180 181 182
|
syl2anc2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) → 𝑋 ≠ ∅ ) |
184 |
122 179 183
|
pm2.61ne |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) → 𝑋 ≠ ∪ 𝑦 ) |
185 |
184
|
neneqd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) ) → ¬ 𝑋 = ∪ 𝑦 ) |
186 |
185
|
nrexdv |
⊢ ( 𝜑 → ¬ ∃ 𝑦 ∈ ( 𝒫 ( 𝐵 ∖ 𝐹 ) ∩ Fin ) 𝑋 = ∪ 𝑦 ) |
187 |
118 186
|
pm2.65i |
⊢ ¬ 𝜑 |