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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 487 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 ⊆ 𝑌) | |
2 | sseqin2 4192 | . . . . 5 ⊢ (𝐴 ⊆ 𝑌 ↔ (𝑌 ∩ 𝐴) = 𝐴) | |
3 | 1, 2 | sylib 220 | . . . 4 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (𝑌 ∩ 𝐴) = 𝐴) |
4 | simpl 485 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐿 ∈ (Fil‘𝑌)) | |
5 | id 22 | . . . . . 6 ⊢ (𝐴 ⊆ 𝑌 → 𝐴 ⊆ 𝑌) | |
6 | filtop 22463 | . . . . . 6 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝑌 ∈ 𝐿) | |
7 | ssexg 5227 | . . . . . 6 ⊢ ((𝐴 ⊆ 𝑌 ∧ 𝑌 ∈ 𝐿) → 𝐴 ∈ V) | |
8 | 5, 6, 7 | syl2anr 598 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 ∈ V) |
9 | 6 | adantr 483 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝑌 ∈ 𝐿) |
10 | elrestr 16702 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ∈ V ∧ 𝑌 ∈ 𝐿) → (𝑌 ∩ 𝐴) ∈ (𝐿 ↾t 𝐴)) | |
11 | 4, 8, 9, 10 | syl3anc 1367 | . . . 4 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (𝑌 ∩ 𝐴) ∈ (𝐿 ↾t 𝐴)) |
12 | 3, 11 | eqeltrrd 2914 | . . 3 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 ∈ (𝐿 ↾t 𝐴)) |
13 | elssuni 4868 | . . 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 16705 | . . . 4 ⊢ (𝐿 ↾t 𝐴) ⊆ 𝒫 𝐴 | |
16 | sspwuni 5022 | . . . 4 ⊢ ((𝐿 ↾t 𝐴) ⊆ 𝒫 𝐴 ↔ ∪ (𝐿 ↾t 𝐴) ⊆ 𝐴) | |
17 | 15, 16 | mpbi 232 | . . 3 ⊢ ∪ (𝐿 ↾t 𝐴) ⊆ 𝐴 |
18 | 17 | a1i 11 | . 2 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ∪ (𝐿 ↾t 𝐴) ⊆ 𝐴) |
19 | 14, 18 | eqssd 3984 | 1 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → 𝐴 = ∪ (𝐿 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4838 ‘cfv 6355 (class class class)co 7156 ↾t crest 16694 Filcfil 22453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-rest 16696 df-fbas 20542 df-fil 22454 |
This theorem is referenced by: (None) |
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