Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 21461 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹))) | |
2 | 1 | simprbi 479 | . . 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 6130 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ dom Fil) | |
5 | simp2 1055 | . . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵) → 𝐵 ⊆ 𝑋) | |
6 | elpw2g 4754 | . . . 4 ⊢ (𝑋 ∈ dom Fil → (𝐵 ∈ 𝒫 𝑋 ↔ 𝐵 ⊆ 𝑋)) | |
7 | 6 | biimpar 501 | . . 3 ⊢ ((𝑋 ∈ dom Fil ∧ 𝐵 ⊆ 𝑋) → 𝐵 ∈ 𝒫 𝑋) |
8 | 4, 5, 7 | syl2an 493 | . 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → 𝐵 ∈ 𝒫 𝑋) |
9 | simpr1 1060 | . . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → 𝐴 ∈ 𝐹) | |
10 | simpr3 1062 | . . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → 𝐴 ⊆ 𝐵) | |
11 | elpwg 4116 | . . . . 5 ⊢ (𝐴 ∈ 𝐹 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) | |
12 | 9, 11 | syl 17 | . . . 4 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)) |
13 | 10, 12 | mpbird 246 | . . 3 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → 𝐴 ∈ 𝒫 𝐵) |
14 | inelcm 3984 | . . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐴 ∈ 𝒫 𝐵) → (𝐹 ∩ 𝒫 𝐵) ≠ ∅) | |
15 | 9, 13, 14 | syl2anc 691 | . 2 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → (𝐹 ∩ 𝒫 𝐵) ≠ ∅) |
16 | pweq 4111 | . . . . . 6 ⊢ (𝑥 = 𝐵 → 𝒫 𝑥 = 𝒫 𝐵) | |
17 | 16 | ineq2d 3776 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝐹 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝐵)) |
18 | 17 | neeq1d 2841 | . . . 4 ⊢ (𝑥 = 𝐵 → ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝐵) ≠ ∅)) |
19 | eleq1 2676 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐹 ↔ 𝐵 ∈ 𝐹)) | |
20 | 18, 19 | imbi12d 333 | . . 3 ⊢ (𝑥 = 𝐵 → (((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹) ↔ ((𝐹 ∩ 𝒫 𝐵) ≠ ∅ → 𝐵 ∈ 𝐹))) |
21 | 20 | rspccv 3279 | . 2 ⊢ (∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹) → (𝐵 ∈ 𝒫 𝑋 → ((𝐹 ∩ 𝒫 𝐵) ≠ ∅ → 𝐵 ∈ 𝐹))) |
22 | 3, 8, 15, 21 | syl3c 64 | 1 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → 𝐵 ∈ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 dom cdm 5038 ‘cfv 5804 fBascfbas 19555 Filcfil 21459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-fil 21460 |
This theorem is referenced by: filin 21468 filtop 21469 isfil2 21470 infil 21477 fgfil 21489 fgabs 21493 filcon 21497 filuni 21499 trfil2 21501 trfg 21505 isufil2 21522 ufprim 21523 ufileu 21533 filufint 21534 elfm3 21564 rnelfm 21567 fmfnfmlem2 21569 fmfnfmlem4 21571 flimopn 21589 flimrest 21597 flimfnfcls 21642 fclscmpi 21643 alexsublem 21658 metust 22173 cfil3i 22875 cfilfcls 22880 iscmet3lem2 22898 equivcfil 22905 relcmpcmet 22923 minveclem4 23011 fgmin 31535 |
Copyright terms: Public domain | W3C validator |