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Mirrors > Home > MPE Home > Th. List > trnei | Structured version Visualization version GIF version |
Description: The trace, over a set 𝐴, of the filter of the neighborhoods of a point 𝑃 is a filter iff 𝑃 belongs to the closure of 𝐴. (This is trfil2 23911 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
Ref | Expression |
---|---|
trnei | ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 22935 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑌) → 𝐽 ∈ Top) | |
2 | 1 | 3ad2ant1 1132 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐽 ∈ Top) |
3 | simp2 1136 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐴 ⊆ 𝑌) | |
4 | toponuni 22936 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐽) | |
5 | 4 | 3ad2ant1 1132 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝑌 = ∪ 𝐽) |
6 | 3, 5 | sseqtrd 4036 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐴 ⊆ ∪ 𝐽) |
7 | simp3 1137 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝑃 ∈ 𝑌) | |
8 | 7, 5 | eleqtrd 2841 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝑃 ∈ ∪ 𝐽) |
9 | eqid 2735 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
10 | 9 | neindisj2 23147 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑃 ∈ ∪ 𝐽) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣 ∩ 𝐴) ≠ ∅)) |
11 | 2, 6, 8, 10 | syl3anc 1370 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣 ∩ 𝐴) ≠ ∅)) |
12 | simp1 1135 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → 𝐽 ∈ (TopOn‘𝑌)) | |
13 | 7 | snssd 4814 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → {𝑃} ⊆ 𝑌) |
14 | snnzg 4779 | . . . . 5 ⊢ (𝑃 ∈ 𝑌 → {𝑃} ≠ ∅) | |
15 | 14 | 3ad2ant3 1134 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → {𝑃} ≠ ∅) |
16 | neifil 23904 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ {𝑃} ⊆ 𝑌 ∧ {𝑃} ≠ ∅) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌)) | |
17 | 12, 13, 15, 16 | syl3anc 1370 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → ((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌)) |
18 | trfil2 23911 | . . 3 ⊢ ((((nei‘𝐽)‘{𝑃}) ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣 ∩ 𝐴) ≠ ∅)) | |
19 | 17, 3, 18 | syl2anc 584 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → ((((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑃})(𝑣 ∩ 𝐴) ≠ ∅)) |
20 | 11, 19 | bitr4d 282 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 {csn 4631 ∪ cuni 4912 ‘cfv 6563 (class class class)co 7431 ↾t crest 17467 Topctop 22915 TopOnctopon 22932 clsccl 23042 neicnei 23121 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-int 4952 df-iun 4998 df-iin 4999 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-top 22916 df-topon 22933 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-fil 23870 |
This theorem is referenced by: flfcntr 24067 cnextfun 24088 cnextfvval 24089 cnextf 24090 cnextcn 24091 cnextfres1 24092 cnextucn 24328 ucnextcn 24329 limcflflem 25930 rrhre 33984 |
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