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Mirrors > Home > MPE Home > Th. List > flimtopon | Structured version Visualization version GIF version |
Description: Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
flimtopon | ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flimtop 23451 | . . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top) | |
2 | istopon 22396 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) ↔ (𝐽 ∈ Top ∧ 𝑋 = ∪ 𝐽)) | |
3 | 2 | baib 537 | . . 3 ⊢ (𝐽 ∈ Top → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = ∪ 𝐽)) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝑋 = ∪ 𝐽)) |
5 | eqid 2733 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
6 | 5 | flimfil 23455 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐽)) |
7 | fveq2 6888 | . . . . 5 ⊢ (𝑋 = ∪ 𝐽 → (Fil‘𝑋) = (Fil‘∪ 𝐽)) | |
8 | 7 | eleq2d 2820 | . . . 4 ⊢ (𝑋 = ∪ 𝐽 → (𝐹 ∈ (Fil‘𝑋) ↔ 𝐹 ∈ (Fil‘∪ 𝐽))) |
9 | 6, 8 | syl5ibrcom 246 | . . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑋 = ∪ 𝐽 → 𝐹 ∈ (Fil‘𝑋))) |
10 | filunibas 23367 | . . . . 5 ⊢ (𝐹 ∈ (Fil‘∪ 𝐽) → ∪ 𝐹 = ∪ 𝐽) | |
11 | 6, 10 | syl 17 | . . . 4 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ∪ 𝐹 = ∪ 𝐽) |
12 | filunibas 23367 | . . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋) | |
13 | 12 | eqeq1d 2735 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∪ 𝐹 = ∪ 𝐽 ↔ 𝑋 = ∪ 𝐽)) |
14 | 11, 13 | syl5ibcom 244 | . . 3 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐹 ∈ (Fil‘𝑋) → 𝑋 = ∪ 𝐽)) |
15 | 9, 14 | impbid 211 | . 2 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝑋 = ∪ 𝐽 ↔ 𝐹 ∈ (Fil‘𝑋))) |
16 | 4, 15 | bitrd 279 | 1 ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∪ cuni 4907 ‘cfv 6540 (class class class)co 7404 Topctop 22377 TopOnctopon 22394 Filcfil 23331 fLim cflim 23420 |
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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-fbas 20926 df-top 22378 df-topon 22395 df-nei 22584 df-fil 23332 df-flim 23425 |
This theorem is referenced by: fclsfnflim 23513 |
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