Step | Hyp | Ref
| Expression |
1 | | flimfcls 21640 |
. . . . 5
⊢ (𝐽 fLim 𝑔) ⊆ (𝐽 fClus 𝑔) |
2 | | flimtop 21579 |
. . . . . . . . 9
⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top) |
3 | | flimfnfcls.x |
. . . . . . . . . 10
⊢ 𝑋 = ∪
𝐽 |
4 | 3 | toptopon 20548 |
. . . . . . . . 9
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
5 | 2, 4 | sylib 207 |
. . . . . . . 8
⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ (TopOn‘𝑋)) |
6 | 5 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐽 ∈ (TopOn‘𝑋)) |
7 | | simplr 788 |
. . . . . . 7
⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝑔 ∈ (Fil‘𝑋)) |
8 | | simpr 476 |
. . . . . . 7
⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐹 ⊆ 𝑔) |
9 | | flimss2 21586 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑔 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝑔) → (𝐽 fLim 𝐹) ⊆ (𝐽 fLim 𝑔)) |
10 | 6, 7, 8, 9 | syl3anc 1318 |
. . . . . 6
⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → (𝐽 fLim 𝐹) ⊆ (𝐽 fLim 𝑔)) |
11 | | simpll 786 |
. . . . . 6
⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐴 ∈ (𝐽 fLim 𝐹)) |
12 | 10, 11 | sseldd 3569 |
. . . . 5
⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐴 ∈ (𝐽 fLim 𝑔)) |
13 | 1, 12 | sseldi 3566 |
. . . 4
⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) ∧ 𝐹 ⊆ 𝑔) → 𝐴 ∈ (𝐽 fClus 𝑔)) |
14 | 13 | ex 449 |
. . 3
⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑔 ∈ (Fil‘𝑋)) → (𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔))) |
15 | 14 | ralrimiva 2949 |
. 2
⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔))) |
16 | | sseq2 3590 |
. . . . . 6
⊢ (𝑔 = 𝐹 → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ 𝐹)) |
17 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑔 = 𝐹 → (𝐽 fClus 𝑔) = (𝐽 fClus 𝐹)) |
18 | 17 | eleq2d 2673 |
. . . . . 6
⊢ (𝑔 = 𝐹 → (𝐴 ∈ (𝐽 fClus 𝑔) ↔ 𝐴 ∈ (𝐽 fClus 𝐹))) |
19 | 16, 18 | imbi12d 333 |
. . . . 5
⊢ (𝑔 = 𝐹 → ((𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) ↔ (𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹)))) |
20 | 19 | rspcv 3278 |
. . . 4
⊢ (𝐹 ∈ (Fil‘𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹)))) |
21 | | ssid 3587 |
. . . . . 6
⊢ 𝐹 ⊆ 𝐹 |
22 | | id 22 |
. . . . . 6
⊢ ((𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹))) |
23 | 21, 22 | mpi 20 |
. . . . 5
⊢ ((𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹)) → 𝐴 ∈ (𝐽 fClus 𝐹)) |
24 | | fclstop 21625 |
. . . . . 6
⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top) |
25 | 3 | fclselbas 21630 |
. . . . . 6
⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 ∈ 𝑋) |
26 | 24, 25 | jca 553 |
. . . . 5
⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) |
27 | 23, 26 | syl 17 |
. . . 4
⊢ ((𝐹 ⊆ 𝐹 → 𝐴 ∈ (𝐽 fClus 𝐹)) → (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) |
28 | 20, 27 | syl6 34 |
. . 3
⊢ (𝐹 ∈ (Fil‘𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋))) |
29 | | disjdif 3992 |
. . . . . . . . . . . . . 14
⊢ (𝑜 ∩ (𝑋 ∖ 𝑜)) = ∅ |
30 | | simpll 786 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝐹 ∈ (Fil‘𝑋)) |
31 | | simplrl 796 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝐽 ∈ Top) |
32 | 3 | topopn 20536 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
33 | 31, 32 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝑋 ∈ 𝐽) |
34 | | pwexg 4776 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V) |
35 | | rabexg 4739 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(𝒫 𝑋 ∈
V → {𝑥 ∈
𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ V) |
36 | 33, 34, 35 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ V) |
37 | | unexg 6857 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ V) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ∈ V) |
38 | 30, 36, 37 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ∈ V) |
39 | | ssfii 8208 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ∈ V → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) |
40 | 38, 39 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) |
41 | | filsspw 21465 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋) |
42 | | ssrab2 3650 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ⊆ 𝒫 𝑋 |
43 | 42 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐹 ∈ (Fil‘𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ⊆ 𝒫 𝑋) |
44 | 41, 43 | unssd 3751 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ 𝒫 𝑋) |
45 | 44 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ 𝒫 𝑋) |
46 | | ssun2 3739 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ⊆ (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) |
47 | | difss 3699 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑋 ∖ 𝑜) ⊆ 𝑋 |
48 | | elpw2g 4754 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑋 ∈ 𝐽 → ((𝑋 ∖ 𝑜) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑜) ⊆ 𝑋)) |
49 | 33, 48 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ((𝑋 ∖ 𝑜) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑜) ⊆ 𝑋)) |
50 | 47, 49 | mpbiri 247 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ∈ 𝒫 𝑋) |
51 | | ssid 3587 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑋 ∖ 𝑜) ⊆ (𝑋 ∖ 𝑜) |
52 | 51 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ⊆ (𝑋 ∖ 𝑜)) |
53 | | sseq2 3590 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = (𝑋 ∖ 𝑜) → ((𝑋 ∖ 𝑜) ⊆ 𝑥 ↔ (𝑋 ∖ 𝑜) ⊆ (𝑋 ∖ 𝑜))) |
54 | 53 | elrab 3331 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑋 ∖ 𝑜) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ↔ ((𝑋 ∖ 𝑜) ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑜) ⊆ (𝑋 ∖ 𝑜))) |
55 | 50, 52, 54 | sylanbrc 695 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) |
56 | 46, 55 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ∈ (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) |
57 | | ne0i 3880 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑋 ∖ 𝑜) ∈ (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ≠ ∅) |
58 | 56, 57 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ≠ ∅) |
59 | | sseq2 3590 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑥 = 𝑧 → ((𝑋 ∖ 𝑜) ⊆ 𝑥 ↔ (𝑋 ∖ 𝑜) ⊆ 𝑧)) |
60 | 59 | elrab 3331 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ↔ (𝑧 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑜) ⊆ 𝑧)) |
61 | 60 | simprbi 479 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} → (𝑋 ∖ 𝑜) ⊆ 𝑧) |
62 | 61 | ad2antll 761 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → (𝑋 ∖ 𝑜) ⊆ 𝑧) |
63 | | sslin 3801 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑋 ∖ 𝑜) ⊆ 𝑧 → (𝑦 ∩ (𝑋 ∖ 𝑜)) ⊆ (𝑦 ∩ 𝑧)) |
64 | 62, 63 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → (𝑦 ∩ (𝑋 ∖ 𝑜)) ⊆ (𝑦 ∩ 𝑧)) |
65 | | simprrr 801 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ¬ 𝑜 ∈ 𝐹) |
66 | 65 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → ¬ 𝑜 ∈ 𝐹) |
67 | | inssdif0 3901 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑦 ∩ 𝑋) ⊆ 𝑜 ↔ (𝑦 ∩ (𝑋 ∖ 𝑜)) = ∅) |
68 | | simplll 794 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → 𝐹 ∈ (Fil‘𝑋)) |
69 | | simprl 790 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → 𝑦 ∈ 𝐹) |
70 | | filelss 21466 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑦 ∈ 𝐹) → 𝑦 ⊆ 𝑋) |
71 | 68, 69, 70 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → 𝑦 ⊆ 𝑋) |
72 | | df-ss 3554 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑦 ⊆ 𝑋 ↔ (𝑦 ∩ 𝑋) = 𝑦) |
73 | 71, 72 | sylib 207 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → (𝑦 ∩ 𝑋) = 𝑦) |
74 | 73 | sseq1d 3595 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → ((𝑦 ∩ 𝑋) ⊆ 𝑜 ↔ 𝑦 ⊆ 𝑜)) |
75 | 30 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝐹 ∈
(Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∧ 𝑦 ⊆ 𝑜) → 𝐹 ∈ (Fil‘𝑋)) |
76 | | simplrl 796 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝐹 ∈
(Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∧ 𝑦 ⊆ 𝑜) → 𝑦 ∈ 𝐹) |
77 | | elssuni 4403 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ ∪ 𝐽) |
78 | 77, 3 | syl6sseqr 3615 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑜 ∈ 𝐽 → 𝑜 ⊆ 𝑋) |
79 | 78 | ad2antrl 760 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝑜 ⊆ 𝑋) |
80 | 79 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝐹 ∈
(Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∧ 𝑦 ⊆ 𝑜) → 𝑜 ⊆ 𝑋) |
81 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((𝐹 ∈
(Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∧ 𝑦 ⊆ 𝑜) → 𝑦 ⊆ 𝑜) |
82 | | filss 21467 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑦 ∈ 𝐹 ∧ 𝑜 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑜)) → 𝑜 ∈ 𝐹) |
83 | 75, 76, 80, 81, 82 | syl13anc 1320 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(((((𝐹 ∈
(Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∧ 𝑦 ⊆ 𝑜) → 𝑜 ∈ 𝐹) |
84 | 83 | ex 449 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → (𝑦 ⊆ 𝑜 → 𝑜 ∈ 𝐹)) |
85 | 74, 84 | sylbid 229 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → ((𝑦 ∩ 𝑋) ⊆ 𝑜 → 𝑜 ∈ 𝐹)) |
86 | 67, 85 | syl5bir 232 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → ((𝑦 ∩ (𝑋 ∖ 𝑜)) = ∅ → 𝑜 ∈ 𝐹)) |
87 | 86 | necon3bd 2796 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → (¬ 𝑜 ∈ 𝐹 → (𝑦 ∩ (𝑋 ∖ 𝑜)) ≠ ∅)) |
88 | 66, 87 | mpd 15 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → (𝑦 ∩ (𝑋 ∖ 𝑜)) ≠ ∅) |
89 | | ssn0 3928 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑦 ∩ (𝑋 ∖ 𝑜)) ⊆ (𝑦 ∩ 𝑧) ∧ (𝑦 ∩ (𝑋 ∖ 𝑜)) ≠ ∅) → (𝑦 ∩ 𝑧) ≠ ∅) |
90 | 64, 88, 89 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ (𝑦 ∈ 𝐹 ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) → (𝑦 ∩ 𝑧) ≠ ∅) |
91 | 90 | ralrimivva 2954 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} (𝑦 ∩ 𝑧) ≠ ∅) |
92 | | filfbas 21462 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) |
93 | 30, 92 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝐹 ∈ (fBas‘𝑋)) |
94 | 47 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ⊆ 𝑋) |
95 | | filtop 21469 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹) |
96 | 30, 95 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝑋 ∈ 𝐹) |
97 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑜 = 𝑋 → (𝑜 ∈ 𝐹 ↔ 𝑋 ∈ 𝐹)) |
98 | 96, 97 | syl5ibrcom 236 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑜 = 𝑋 → 𝑜 ∈ 𝐹)) |
99 | 98 | necon3bd 2796 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (¬ 𝑜 ∈ 𝐹 → 𝑜 ≠ 𝑋)) |
100 | 65, 99 | mpd 15 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝑜 ≠ 𝑋) |
101 | | pssdifn0 3898 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑜 ⊆ 𝑋 ∧ 𝑜 ≠ 𝑋) → (𝑋 ∖ 𝑜) ≠ ∅) |
102 | 79, 100, 101 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ≠ ∅) |
103 | | supfil 21509 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑋 ∈ 𝐽 ∧ (𝑋 ∖ 𝑜) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑜) ≠ ∅) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (Fil‘𝑋)) |
104 | 33, 94, 102, 103 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (Fil‘𝑋)) |
105 | | filfbas 21462 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ({𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (Fil‘𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (fBas‘𝑋)) |
106 | 104, 105 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (fBas‘𝑋)) |
107 | | fbunfip 21483 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} ∈ (fBas‘𝑋)) → (¬ ∅ ∈
(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ↔ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} (𝑦 ∩ 𝑧) ≠ ∅)) |
108 | 93, 106, 107 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (¬ ∅ ∈
(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ↔ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥} (𝑦 ∩ 𝑧) ≠ ∅)) |
109 | 91, 108 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ¬ ∅ ∈
(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) |
110 | | fsubbas 21481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑋 ∈ 𝐹 → ((fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∈ (fBas‘𝑋) ↔ ((𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ 𝒫 𝑋 ∧ (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ≠ ∅ ∧ ¬ ∅ ∈
(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) |
111 | 96, 110 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ((fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∈ (fBas‘𝑋) ↔ ((𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ 𝒫 𝑋 ∧ (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ≠ ∅ ∧ ¬ ∅ ∈
(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) |
112 | 45, 58, 109, 111 | mpbir3and 1238 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∈ (fBas‘𝑋)) |
113 | | ssfg 21486 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((fi‘(𝐹 ∪
{𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∈ (fBas‘𝑋) → (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))) |
114 | 112, 113 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))) |
115 | 40, 114 | sstrd 3578 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}) ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))) |
116 | 115 | unssad 3752 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))) |
117 | | fgcl 21492 |
. . . . . . . . . . . . . . . . . . 19
⊢
((fi‘(𝐹 ∪
{𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) ∈ (Fil‘𝑋)) |
118 | 112, 117 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) ∈ (Fil‘𝑋)) |
119 | | sseq2 3590 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) → (𝐹 ⊆ 𝑔 ↔ 𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) |
120 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) → (𝐽 fClus 𝑔) = (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) |
121 | 120 | eleq2d 2673 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) → (𝐴 ∈ (𝐽 fClus 𝑔) ↔ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))))) |
122 | 119, 121 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑔 = (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) → ((𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) ↔ (𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) → 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))))) |
123 | 122 | rspcv 3278 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) ∈ (Fil‘𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) → 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))))) |
124 | 118, 123 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐹 ⊆ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))) → 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))))) |
125 | 116, 124 | mpid 43 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))))) |
126 | | simpr 476 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) → 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) |
127 | | simplrl 796 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) → 𝑜 ∈ 𝐽) |
128 | | simprrl 800 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → 𝐴 ∈ 𝑜) |
129 | 128 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) → 𝐴 ∈ 𝑜) |
130 | 115, 56 | sseldd 3569 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝑋 ∖ 𝑜) ∈ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))) |
131 | 130 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) → (𝑋 ∖ 𝑜) ∈ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))) |
132 | | fclsopni 21629 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜 ∧ (𝑋 ∖ 𝑜) ∈ (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) → (𝑜 ∩ (𝑋 ∖ 𝑜)) ≠ ∅) |
133 | 126, 127,
129, 131, 132 | syl13anc 1320 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) ∧ 𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥}))))) → (𝑜 ∩ (𝑋 ∖ 𝑜)) ≠ ∅) |
134 | 133 | ex 449 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (𝐴 ∈ (𝐽 fClus (𝑋filGen(fi‘(𝐹 ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑜) ⊆ 𝑥})))) → (𝑜 ∩ (𝑋 ∖ 𝑜)) ≠ ∅)) |
135 | 125, 134 | syld 46 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝑜 ∩ (𝑋 ∖ 𝑜)) ≠ ∅)) |
136 | 135 | necon2bd 2798 |
. . . . . . . . . . . . . 14
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ((𝑜 ∩ (𝑋 ∖ 𝑜)) = ∅ → ¬ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)))) |
137 | 29, 136 | mpi 20 |
. . . . . . . . . . . . 13
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ (𝑜 ∈ 𝐽 ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹))) → ¬ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔))) |
138 | 137 | anassrs 678 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ 𝑜 ∈ 𝐽) ∧ (𝐴 ∈ 𝑜 ∧ ¬ 𝑜 ∈ 𝐹)) → ¬ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔))) |
139 | 138 | expr 641 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ 𝑜 ∈ 𝐽) ∧ 𝐴 ∈ 𝑜) → (¬ 𝑜 ∈ 𝐹 → ¬ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)))) |
140 | 139 | con4d 113 |
. . . . . . . . . 10
⊢ ((((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ 𝑜 ∈ 𝐽) ∧ 𝐴 ∈ 𝑜) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → 𝑜 ∈ 𝐹)) |
141 | 140 | ex 449 |
. . . . . . . . 9
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ 𝑜 ∈ 𝐽) → (𝐴 ∈ 𝑜 → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → 𝑜 ∈ 𝐹))) |
142 | 141 | com23 84 |
. . . . . . . 8
⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) ∧ 𝑜 ∈ 𝐽) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐴 ∈ 𝑜 → 𝑜 ∈ 𝐹))) |
143 | 142 | ralrimdva 2952 |
. . . . . . 7
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑜 ∈ 𝐹))) |
144 | | simprr 792 |
. . . . . . 7
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋) |
145 | 143, 144 | jctild 564 |
. . . . . 6
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑜 ∈ 𝐹)))) |
146 | | simprl 790 |
. . . . . . . 8
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → 𝐽 ∈ Top) |
147 | 146, 4 | sylib 207 |
. . . . . . 7
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → 𝐽 ∈ (TopOn‘𝑋)) |
148 | | simpl 472 |
. . . . . . 7
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → 𝐹 ∈ (Fil‘𝑋)) |
149 | | flimopn 21589 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑜 ∈ 𝐹)))) |
150 | 147, 148,
149 | syl2anc 691 |
. . . . . 6
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑜 ∈ 𝐹)))) |
151 | 145, 150 | sylibrd 248 |
. . . . 5
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋)) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → 𝐴 ∈ (𝐽 fLim 𝐹))) |
152 | 151 | ex 449 |
. . . 4
⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → 𝐴 ∈ (𝐽 fLim 𝐹)))) |
153 | 152 | com23 84 |
. . 3
⊢ (𝐹 ∈ (Fil‘𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim 𝐹)))) |
154 | 28, 153 | mpdd 42 |
. 2
⊢ (𝐹 ∈ (Fil‘𝑋) → (∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)) → 𝐴 ∈ (𝐽 fLim 𝐹))) |
155 | 15, 154 | impbid2 215 |
1
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔)))) |