Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > trfil1 | Structured version Visualization version GIF version |
Description: Conditions for the trace of a filter 𝐿 to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
Ref | Expression |
---|---|
trfil1 | ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 = ∪ (𝐿 ↾t 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 ⊆ 𝑌) | |
2 | sseqin2 4189 | . . . . 5 ⊢ (𝐴 ⊆ 𝑌 ↔ (𝑌 ∩ 𝐴) = 𝐴) | |
3 | 1, 2 | sylib 219 | . . . 4 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (𝑌 ∩ 𝐴) = 𝐴) |
4 | simpl 483 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐿 ∈ (Fil‘𝑌)) | |
5 | id 22 | . . . . . 6 ⊢ (𝐴 ⊆ 𝑌 → 𝐴 ⊆ 𝑌) | |
6 | filtop 22391 | . . . . . 6 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝑌 ∈ 𝐿) | |
7 | ssexg 5218 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝑌 ∧ 𝑌 ∈ 𝐿) → 𝐴 ∈ V) | |
8 | 5, 6, 7 | syl2anr 596 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 ∈ V) |
9 | 6 | adantr 481 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝑌 ∈ 𝐿) |
10 | elrestr 16690 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ∈ V ∧ 𝑌 ∈ 𝐿) → (𝑌 ∩ 𝐴) ∈ (𝐿 ↾t 𝐴)) | |
11 | 4, 8, 9, 10 | syl3anc 1363 | . . . 4 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (𝑌 ∩ 𝐴) ∈ (𝐿 ↾t 𝐴)) |
12 | 3, 11 | eqeltrrd 2911 | . . 3 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 ∈ (𝐿 ↾t 𝐴)) |
13 | elssuni 4859 | . . 3 ⊢ (𝐴 ∈ (𝐿 ↾t 𝐴) → 𝐴 ⊆ ∪ (𝐿 ↾t 𝐴)) | |
14 | 12, 13 | syl 17 | . 2 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 ⊆ ∪ (𝐿 ↾t 𝐴)) |
15 | restsspw 16693 | . . . 4 ⊢ (𝐿 ↾t 𝐴) ⊆ 𝒫 𝐴 | |
16 | sspwuni 5013 | . . . 4 ⊢ ((𝐿 ↾t 𝐴) ⊆ 𝒫 𝐴 ↔ ∪ (𝐿 ↾t 𝐴) ⊆ 𝐴) | |
17 | 15, 16 | mpbi 231 | . . 3 ⊢ ∪ (𝐿 ↾t 𝐴) ⊆ 𝐴 |
18 | 17 | a1i 11 | . 2 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ∪ (𝐿 ↾t 𝐴) ⊆ 𝐴) |
19 | 14, 18 | eqssd 3981 | 1 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 = ∪ (𝐿 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 𝒫 cpw 4535 ∪ cuni 4830 ‘cfv 6348 (class class class)co 7145 ↾t crest 16682 Filcfil 22381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-rest 16684 df-fbas 20470 df-fil 22382 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |