Proof of Theorem uffixfr
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpl 482 | . . 3
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 ∈ (UFil‘𝑋)) | 
| 2 |  | ufilfil 23912 | . . . . . . 7
⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋)) | 
| 3 |  | filtop 23863 | . . . . . . 7
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹) | 
| 4 | 2, 3 | syl 17 | . . . . . 6
⊢ (𝐹 ∈ (UFil‘𝑋) → 𝑋 ∈ 𝐹) | 
| 5 |  | filn0 23870 | . . . . . . . . 9
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅) | 
| 6 |  | intssuni 4970 | . . . . . . . . 9
⊢ (𝐹 ≠ ∅ → ∩ 𝐹
⊆ ∪ 𝐹) | 
| 7 | 2, 5, 6 | 3syl 18 | . . . . . . . 8
⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹
⊆ ∪ 𝐹) | 
| 8 |  | filunibas 23889 | . . . . . . . . 9
⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 =
𝑋) | 
| 9 | 2, 8 | syl 17 | . . . . . . . 8
⊢ (𝐹 ∈ (UFil‘𝑋) → ∪ 𝐹 =
𝑋) | 
| 10 | 7, 9 | sseqtrd 4020 | . . . . . . 7
⊢ (𝐹 ∈ (UFil‘𝑋) → ∩ 𝐹
⊆ 𝑋) | 
| 11 | 10 | sselda 3983 | . . . . . 6
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐴 ∈ 𝑋) | 
| 12 |  | uffix 23929 | . . . . . 6
⊢ ((𝑋 ∈ 𝐹 ∧ 𝐴 ∈ 𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}}))) | 
| 13 | 4, 11, 12 | syl2an2r 685 | . . . . 5
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}}))) | 
| 14 | 13 | simprd 495 | . . . 4
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}})) | 
| 15 | 13 | simpld 494 | . . . . 5
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {{𝐴}} ∈ (fBas‘𝑋)) | 
| 16 |  | fgcl 23886 | . . . . 5
⊢ ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋)) | 
| 17 | 15, 16 | syl 17 | . . . 4
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋)) | 
| 18 | 14, 17 | eqeltrd 2841 | . . 3
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (Fil‘𝑋)) | 
| 19 | 2 | adantr 480 | . . . . 5
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 ∈ (Fil‘𝑋)) | 
| 20 |  | filsspw 23859 | . . . . 5
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋) | 
| 21 | 19, 20 | syl 17 | . . . 4
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 ⊆ 𝒫 𝑋) | 
| 22 |  | elintg 4954 | . . . . . 6
⊢ (𝐴 ∈ ∩ 𝐹
→ (𝐴 ∈ ∩ 𝐹
↔ ∀𝑥 ∈
𝐹 𝐴 ∈ 𝑥)) | 
| 23 | 22 | ibi 267 | . . . . 5
⊢ (𝐴 ∈ ∩ 𝐹
→ ∀𝑥 ∈
𝐹 𝐴 ∈ 𝑥) | 
| 24 | 23 | adantl 481 | . . . 4
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥) | 
| 25 |  | ssrab 4073 | . . . 4
⊢ (𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥)) | 
| 26 | 21, 24, 25 | sylanbrc 583 | . . 3
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) | 
| 27 |  | ufilmax 23915 | . . 3
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) | 
| 28 | 1, 18, 26, 27 | syl3anc 1373 | . 2
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ∈ ∩ 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) | 
| 29 |  | eqimss 4042 | . . . . 5
⊢ (𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) | 
| 30 | 29 | adantl 481 | . . . 4
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) | 
| 31 | 25 | simprbi 496 | . . . 4
⊢ (𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} → ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥) | 
| 32 | 30, 31 | syl 17 | . . 3
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥) | 
| 33 |  | eleq2 2830 | . . . . . 6
⊢ (𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} → (𝑋 ∈ 𝐹 ↔ 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})) | 
| 34 | 33 | biimpac 478 | . . . . 5
⊢ ((𝑋 ∈ 𝐹 ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) | 
| 35 | 4, 34 | sylan 580 | . . . 4
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) | 
| 36 |  | eleq2 2830 | . . . . . 6
⊢ (𝑥 = 𝑋 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑋)) | 
| 37 | 36 | elrab 3692 | . . . . 5
⊢ (𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ↔ (𝑋 ∈ 𝒫 𝑋 ∧ 𝐴 ∈ 𝑋)) | 
| 38 | 37 | simprbi 496 | . . . 4
⊢ (𝑋 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} → 𝐴 ∈ 𝑋) | 
| 39 |  | elintg 4954 | . . . 4
⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ ∩ 𝐹 ↔ ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥)) | 
| 40 | 35, 38, 39 | 3syl 18 | . . 3
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → (𝐴 ∈ ∩ 𝐹 ↔ ∀𝑥 ∈ 𝐹 𝐴 ∈ 𝑥)) | 
| 41 | 32, 40 | mpbird 257 | . 2
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}) → 𝐴 ∈ ∩ 𝐹) | 
| 42 | 28, 41 | impbida 801 | 1
⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐴 ∈ ∩ 𝐹 ↔ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥})) |