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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 6678 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ V) | |
2 | uniexg 7446 | . . . 4 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V) | |
3 | eleq1 2877 | . . . 4 ⊢ (𝐵 = ∪ 𝐽 → (𝐵 ∈ V ↔ ∪ 𝐽 ∈ V)) | |
4 | 2, 3 | syl5ibrcom 250 | . . 3 ⊢ (𝐽 ∈ Top → (𝐵 = ∪ 𝐽 → 𝐵 ∈ V)) |
5 | 4 | imp 410 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽) → 𝐵 ∈ V) |
6 | eqeq1 2802 | . . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝑗)) | |
7 | 6 | rabbidv 3427 | . . . . 5 ⊢ (𝑏 = 𝐵 → {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗}) |
8 | df-topon 21516 | . . . . 5 ⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗}) | |
9 | vpwex 5243 | . . . . . . 7 ⊢ 𝒫 𝑏 ∈ V | |
10 | 9 | pwex 5246 | . . . . . 6 ⊢ 𝒫 𝒫 𝑏 ∈ V |
11 | rabss 3999 | . . . . . . 7 ⊢ ({𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏 ↔ ∀𝑗 ∈ Top (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏)) | |
12 | pwuni 4837 | . . . . . . . . . 10 ⊢ 𝑗 ⊆ 𝒫 ∪ 𝑗 | |
13 | pweq 4513 | . . . . . . . . . 10 ⊢ (𝑏 = ∪ 𝑗 → 𝒫 𝑏 = 𝒫 ∪ 𝑗) | |
14 | 12, 13 | sseqtrrid 3968 | . . . . . . . . 9 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ⊆ 𝒫 𝑏) |
15 | velpw 4502 | . . . . . . . . 9 ⊢ (𝑗 ∈ 𝒫 𝒫 𝑏 ↔ 𝑗 ⊆ 𝒫 𝑏) | |
16 | 14, 15 | sylibr 237 | . . . . . . . 8 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏) |
17 | 16 | a1i 11 | . . . . . . 7 ⊢ (𝑗 ∈ Top → (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏)) |
18 | 11, 17 | mprgbir 3121 | . . . . . 6 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏 |
19 | 10, 18 | ssexi 5190 | . . . . 5 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ∈ V |
20 | 7, 8, 19 | fvmpt3i 6750 | . . . 4 ⊢ (𝐵 ∈ V → (TopOn‘𝐵) = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗}) |
21 | 20 | eleq2d 2875 | . . 3 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗})) |
22 | unieq 4811 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
23 | 22 | eqeq2d 2809 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝐵 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝐽)) |
24 | 23 | elrab 3628 | . . 3 ⊢ (𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗} ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)) |
25 | 21, 24 | syl6bb 290 | . 2 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))) |
26 | 1, 5, 25 | pm5.21nii 383 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 ‘cfv 6324 Topctop 21498 TopOnctopon 21515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-topon 21516 |
This theorem is referenced by: topontop 21518 toponuni 21519 toptopon 21522 toponcom 21533 istps2 21540 tgtopon 21576 distopon 21602 indistopon 21606 fctop 21609 cctop 21611 ppttop 21612 epttop 21614 mretopd 21697 toponmre 21698 resttopon 21766 resttopon2 21773 kgentopon 22143 txtopon 22196 pttopon 22201 xkotopon 22205 qtoptopon 22309 flimtopon 22575 fclstopon 22617 fclsfnflim 22632 utoptopon 22842 qtopt1 31188 neibastop1 33820 onsuctopon 33895 rfcnpre1 41648 cnfex 41657 icccncfext 42529 stoweidlem47 42689 |
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