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Mirrors > Home > MPE Home > Th. List > istopon | Structured version Visualization version GIF version |
Description: Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
istopon | ⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6944 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ V) | |
2 | uniexg 7758 | . . . 4 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V) | |
3 | eleq1 2826 | . . . 4 ⊢ (𝐵 = ∪ 𝐽 → (𝐵 ∈ V ↔ ∪ 𝐽 ∈ V)) | |
4 | 2, 3 | syl5ibrcom 247 | . . 3 ⊢ (𝐽 ∈ Top → (𝐵 = ∪ 𝐽 → 𝐵 ∈ V)) |
5 | 4 | imp 406 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽) → 𝐵 ∈ V) |
6 | eqeq1 2738 | . . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝑗)) | |
7 | 6 | rabbidv 3440 | . . . . 5 ⊢ (𝑏 = 𝐵 → {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗}) |
8 | df-topon 22932 | . . . . 5 ⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗}) | |
9 | vpwex 5382 | . . . . . . 7 ⊢ 𝒫 𝑏 ∈ V | |
10 | 9 | pwex 5385 | . . . . . 6 ⊢ 𝒫 𝒫 𝑏 ∈ V |
11 | rabss 4081 | . . . . . . 7 ⊢ ({𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏 ↔ ∀𝑗 ∈ Top (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏)) | |
12 | pwuni 4949 | . . . . . . . . . 10 ⊢ 𝑗 ⊆ 𝒫 ∪ 𝑗 | |
13 | pweq 4618 | . . . . . . . . . 10 ⊢ (𝑏 = ∪ 𝑗 → 𝒫 𝑏 = 𝒫 ∪ 𝑗) | |
14 | 12, 13 | sseqtrrid 4048 | . . . . . . . . 9 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ⊆ 𝒫 𝑏) |
15 | velpw 4609 | . . . . . . . . 9 ⊢ (𝑗 ∈ 𝒫 𝒫 𝑏 ↔ 𝑗 ⊆ 𝒫 𝑏) | |
16 | 14, 15 | sylibr 234 | . . . . . . . 8 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏) |
17 | 16 | a1i 11 | . . . . . . 7 ⊢ (𝑗 ∈ Top → (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏)) |
18 | 11, 17 | mprgbir 3065 | . . . . . 6 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏 |
19 | 10, 18 | ssexi 5327 | . . . . 5 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ∈ V |
20 | 7, 8, 19 | fvmpt3i 7020 | . . . 4 ⊢ (𝐵 ∈ V → (TopOn‘𝐵) = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗}) |
21 | 20 | eleq2d 2824 | . . 3 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗})) |
22 | unieq 4922 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
23 | 22 | eqeq2d 2745 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝐵 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝐽)) |
24 | 23 | elrab 3694 | . . 3 ⊢ (𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗} ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)) |
25 | 21, 24 | bitrdi 287 | . 2 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))) |
26 | 1, 5, 25 | pm5.21nii 378 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {crab 3432 Vcvv 3477 ⊆ wss 3962 𝒫 cpw 4604 ∪ cuni 4911 ‘cfv 6562 Topctop 22914 TopOnctopon 22931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-topon 22932 |
This theorem is referenced by: topontop 22934 toponuni 22935 toptopon 22938 toponcom 22949 istps2 22956 tgtopon 22993 distopon 23019 indistopon 23023 fctop 23026 cctop 23028 ppttop 23029 epttop 23031 mretopd 23115 toponmre 23116 resttopon 23184 resttopon2 23191 kgentopon 23561 txtopon 23614 pttopon 23619 xkotopon 23623 qtoptopon 23727 flimtopon 23993 fclstopon 24035 fclsfnflim 24050 utoptopon 24260 qtopt1 33795 neibastop1 36341 onsuctopon 36416 rfcnpre1 44956 cnfex 44965 icccncfext 45842 stoweidlem47 46002 |
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