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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 477 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 ⊆ 𝑌) | |
2 | sseqin2 3817 | . . . . 5 ⊢ (𝐴 ⊆ 𝑌 ↔ (𝑌 ∩ 𝐴) = 𝐴) | |
3 | 1, 2 | sylib 208 | . . . 4 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (𝑌 ∩ 𝐴) = 𝐴) |
4 | simpl 473 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐿 ∈ (Fil‘𝑌)) | |
5 | id 22 | . . . . . 6 ⊢ (𝐴 ⊆ 𝑌 → 𝐴 ⊆ 𝑌) | |
6 | filtop 21659 | . . . . . 6 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝑌 ∈ 𝐿) | |
7 | ssexg 4804 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝑌 ∧ 𝑌 ∈ 𝐿) → 𝐴 ∈ V) | |
8 | 5, 6, 7 | syl2anr 495 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 ∈ V) |
9 | 6 | adantr 481 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝑌 ∈ 𝐿) |
10 | elrestr 16089 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ∈ V ∧ 𝑌 ∈ 𝐿) → (𝑌 ∩ 𝐴) ∈ (𝐿 ↾t 𝐴)) | |
11 | 4, 8, 9, 10 | syl3anc 1326 | . . . 4 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (𝑌 ∩ 𝐴) ∈ (𝐿 ↾t 𝐴)) |
12 | 3, 11 | eqeltrrd 2702 | . . 3 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 ∈ (𝐿 ↾t 𝐴)) |
13 | elssuni 4467 | . . 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 16092 | . . . 4 ⊢ (𝐿 ↾t 𝐴) ⊆ 𝒫 𝐴 | |
16 | sspwuni 4611 | . . . 4 ⊢ ((𝐿 ↾t 𝐴) ⊆ 𝒫 𝐴 ↔ ∪ (𝐿 ↾t 𝐴) ⊆ 𝐴) | |
17 | 15, 16 | mpbi 220 | . . 3 ⊢ ∪ (𝐿 ↾t 𝐴) ⊆ 𝐴 |
18 | 17 | a1i 11 | . 2 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ∪ (𝐿 ↾t 𝐴) ⊆ 𝐴) |
19 | 14, 18 | eqssd 3620 | 1 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 = ∪ (𝐿 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 ∪ cuni 4436 ‘cfv 5888 (class class class)co 6650 ↾t crest 16081 Filcfil 21649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-rest 16083 df-fbas 19743 df-fil 21650 |
This theorem is referenced by: (None) |
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