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Mirrors > Home > MPE Home > Th. List > trfil3 | Structured version Visualization version GIF version |
Description: Conditions for the trace of a filter 𝐿 to be a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.) |
Ref | Expression |
---|---|
trfil3 | ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝐿 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑌 ∖ 𝐴) ∈ 𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trfil2 23911 | . 2 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝐿 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ 𝐿 (𝑣 ∩ 𝐴) ≠ ∅)) | |
2 | dfral2 3097 | . . 3 ⊢ (∀𝑣 ∈ 𝐿 (𝑣 ∩ 𝐴) ≠ ∅ ↔ ¬ ∃𝑣 ∈ 𝐿 ¬ (𝑣 ∩ 𝐴) ≠ ∅) | |
3 | nne 2942 | . . . . . . . 8 ⊢ (¬ (𝑣 ∩ 𝐴) ≠ ∅ ↔ (𝑣 ∩ 𝐴) = ∅) | |
4 | filelss 23876 | . . . . . . . . 9 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝑣 ∈ 𝐿) → 𝑣 ⊆ 𝑌) | |
5 | reldisj 4459 | . . . . . . . . 9 ⊢ (𝑣 ⊆ 𝑌 → ((𝑣 ∩ 𝐴) = ∅ ↔ 𝑣 ⊆ (𝑌 ∖ 𝐴))) | |
6 | 4, 5 | syl 17 | . . . . . . . 8 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝑣 ∈ 𝐿) → ((𝑣 ∩ 𝐴) = ∅ ↔ 𝑣 ⊆ (𝑌 ∖ 𝐴))) |
7 | 3, 6 | bitrid 283 | . . . . . . 7 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝑣 ∈ 𝐿) → (¬ (𝑣 ∩ 𝐴) ≠ ∅ ↔ 𝑣 ⊆ (𝑌 ∖ 𝐴))) |
8 | 7 | rexbidva 3175 | . . . . . 6 ⊢ (𝐿 ∈ (Fil‘𝑌) → (∃𝑣 ∈ 𝐿 ¬ (𝑣 ∩ 𝐴) ≠ ∅ ↔ ∃𝑣 ∈ 𝐿 𝑣 ⊆ (𝑌 ∖ 𝐴))) |
9 | 8 | adantr 480 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (∃𝑣 ∈ 𝐿 ¬ (𝑣 ∩ 𝐴) ≠ ∅ ↔ ∃𝑣 ∈ 𝐿 𝑣 ⊆ (𝑌 ∖ 𝐴))) |
10 | difssd 4147 | . . . . . 6 ⊢ (𝐴 ⊆ 𝑌 → (𝑌 ∖ 𝐴) ⊆ 𝑌) | |
11 | elfilss 23900 | . . . . . 6 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ (𝑌 ∖ 𝐴) ⊆ 𝑌) → ((𝑌 ∖ 𝐴) ∈ 𝐿 ↔ ∃𝑣 ∈ 𝐿 𝑣 ⊆ (𝑌 ∖ 𝐴))) | |
12 | 10, 11 | sylan2 593 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝑌 ∖ 𝐴) ∈ 𝐿 ↔ ∃𝑣 ∈ 𝐿 𝑣 ⊆ (𝑌 ∖ 𝐴))) |
13 | 9, 12 | bitr4d 282 | . . . 4 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (∃𝑣 ∈ 𝐿 ¬ (𝑣 ∩ 𝐴) ≠ ∅ ↔ (𝑌 ∖ 𝐴) ∈ 𝐿)) |
14 | 13 | notbid 318 | . . 3 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (¬ ∃𝑣 ∈ 𝐿 ¬ (𝑣 ∩ 𝐴) ≠ ∅ ↔ ¬ (𝑌 ∖ 𝐴) ∈ 𝐿)) |
15 | 2, 14 | bitrid 283 | . 2 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (∀𝑣 ∈ 𝐿 (𝑣 ∩ 𝐴) ≠ ∅ ↔ ¬ (𝑌 ∖ 𝐴) ∈ 𝐿)) |
16 | 1, 15 | bitrd 279 | 1 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝐿 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑌 ∖ 𝐴) ∈ 𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ∖ cdif 3960 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 ‘cfv 6563 (class class class)co 7431 ↾t crest 17467 Filcfil 23869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-rest 17469 df-fbas 21379 df-fg 21380 df-fil 23870 |
This theorem is referenced by: fgtr 23914 trufil 23934 flimrest 24007 fclsrest 24048 cfilres 25344 relcmpcmet 25366 |
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