![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 23372 | . 2 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝐿 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ 𝐿 (𝑣 ∩ 𝐴) ≠ ∅)) | |
2 | dfral2 3100 | . . 3 ⊢ (∀𝑣 ∈ 𝐿 (𝑣 ∩ 𝐴) ≠ ∅ ↔ ¬ ∃𝑣 ∈ 𝐿 ¬ (𝑣 ∩ 𝐴) ≠ ∅) | |
3 | nne 2945 | . . . . . . . 8 ⊢ (¬ (𝑣 ∩ 𝐴) ≠ ∅ ↔ (𝑣 ∩ 𝐴) = ∅) | |
4 | filelss 23337 | . . . . . . . . 9 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝑣 ∈ 𝐿) → 𝑣 ⊆ 𝑌) | |
5 | reldisj 4449 | . . . . . . . . 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 3177 | . . . . . 6 ⊢ (𝐿 ∈ (Fil‘𝑌) → (∃𝑣 ∈ 𝐿 ¬ (𝑣 ∩ 𝐴) ≠ ∅ ↔ ∃𝑣 ∈ 𝐿 𝑣 ⊆ (𝑌 ∖ 𝐴))) |
9 | 8 | adantr 482 | . . . . 5 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → (∃𝑣 ∈ 𝐿 ¬ (𝑣 ∩ 𝐴) ≠ ∅ ↔ ∃𝑣 ∈ 𝐿 𝑣 ⊆ (𝑌 ∖ 𝐴))) |
10 | difssd 4130 | . . . . . 6 ⊢ (𝐴 ⊆ 𝑌 → (𝑌 ∖ 𝐴) ⊆ 𝑌) | |
11 | elfilss 23361 | . . . . . 6 ⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ (𝑌 ∖ 𝐴) ⊆ 𝑌) → ((𝑌 ∖ 𝐴) ∈ 𝐿 ↔ ∃𝑣 ∈ 𝐿 𝑣 ⊆ (𝑌 ∖ 𝐴))) | |
12 | 10, 11 | sylan2 594 | . . . . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ∖ cdif 3943 ∩ cin 3945 ⊆ wss 3946 ∅c0 4320 ‘cfv 6539 (class class class)co 7403 ↾t crest 17361 Filcfil 23330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7406 df-oprab 7407 df-mpo 7408 df-1st 7969 df-2nd 7970 df-rest 17363 df-fbas 20925 df-fg 20926 df-fil 23331 |
This theorem is referenced by: fgtr 23375 trufil 23395 flimrest 23468 fclsrest 23509 cfilres 24794 relcmpcmet 24816 |
Copyright terms: Public domain | W3C validator |