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Mirrors > Home > MPE Home > Th. List > filss | Structured version Visualization version GIF version |
Description: A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
Ref | Expression |
---|---|
filss | ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → 𝐵 ∈ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfil 23695 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))) | |
2 | 1 | simprbi 496 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹)) |
3 | 2 | adantr 480 | . 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹)) |
4 | elfvdm 6919 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ dom Fil) | |
5 | simp2 1134 | . . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵) → 𝐵 ⊆ 𝑋) | |
6 | elpw2g 5335 | . . . 4 ⊢ (𝑋 ∈ dom Fil → (𝐵 ∈ 𝒫 𝑋 ↔ 𝐵 ⊆ 𝑋)) | |
7 | 6 | biimpar 477 | . . 3 ⊢ ((𝑋 ∈ dom Fil ∧ 𝐵 ⊆ 𝑋) → 𝐵 ∈ 𝒫 𝑋) |
8 | 4, 5, 7 | syl2an 595 | . 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → 𝐵 ∈ 𝒫 𝑋) |
9 | simpr1 1191 | . . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → 𝐴 ∈ 𝐹) | |
10 | simpr3 1193 | . . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → 𝐴 ⊆ 𝐵) | |
11 | 9, 10 | elpwd 4601 | . . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → 𝐴 ∈ 𝒫 𝐵) |
12 | inelcm 4457 | . . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐴 ∈ 𝒫 𝐵) → (𝐹 ∩ 𝒫 𝐵) ≠ ∅) | |
13 | 9, 11, 12 | syl2anc 583 | . 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → (𝐹 ∩ 𝒫 𝐵) ≠ ∅) |
14 | pweq 4609 | . . . . . 6 ⊢ (𝑥 = 𝐵 → 𝒫 𝑥 = 𝒫 𝐵) | |
15 | 14 | ineq2d 4205 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐹 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝐵)) |
16 | 15 | neeq1d 2992 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝐵) ≠ ∅)) |
17 | eleq1 2813 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐹 ↔ 𝐵 ∈ 𝐹)) | |
18 | 16, 17 | imbi12d 344 | . . 3 ⊢ (𝑥 = 𝐵 → (((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹) ↔ ((𝐹 ∩ 𝒫 𝐵) ≠ ∅ → 𝐵 ∈ 𝐹))) |
19 | 18 | rspccv 3601 | . 2 ⊢ (∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹) → (𝐵 ∈ 𝒫 𝑋 → ((𝐹 ∩ 𝒫 𝐵) ≠ ∅ → 𝐵 ∈ 𝐹))) |
20 | 3, 8, 13, 19 | syl3c 66 | 1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → 𝐵 ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ∩ cin 3940 ⊆ wss 3941 ∅c0 4315 𝒫 cpw 4595 dom cdm 5667 ‘cfv 6534 fBascfbas 21222 Filcfil 23693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fv 6542 df-fil 23694 |
This theorem is referenced by: filin 23702 filtop 23703 isfil2 23704 infil 23711 fgfil 23723 fgabs 23727 filconn 23731 filuni 23733 trfil2 23735 trfg 23739 isufil2 23756 ufprim 23757 ufileu 23767 filufint 23768 elfm3 23798 rnelfm 23801 fmfnfmlem2 23803 fmfnfmlem4 23805 flimopn 23823 flimrest 23831 flimfnfcls 23876 fclscmpi 23877 alexsublem 23892 metust 24411 cfil3i 25141 cfilfcls 25146 iscmet3lem2 25164 equivcfil 25171 relcmpcmet 25190 minveclem4 25304 fgmin 35756 |
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