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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 6705 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ V) | |
2 | uniexg 7468 | . . . 4 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V) | |
3 | eleq1 2902 | . . . 4 ⊢ (𝐵 = ∪ 𝐽 → (𝐵 ∈ V ↔ ∪ 𝐽 ∈ V)) | |
4 | 2, 3 | syl5ibrcom 249 | . . 3 ⊢ (𝐽 ∈ Top → (𝐵 = ∪ 𝐽 → 𝐵 ∈ V)) |
5 | 4 | imp 409 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽) → 𝐵 ∈ V) |
6 | eqeq1 2827 | . . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝑗)) | |
7 | 6 | rabbidv 3482 | . . . . 5 ⊢ (𝑏 = 𝐵 → {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗}) |
8 | df-topon 21521 | . . . . 5 ⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗}) | |
9 | vpwex 5280 | . . . . . . 7 ⊢ 𝒫 𝑏 ∈ V | |
10 | 9 | pwex 5283 | . . . . . 6 ⊢ 𝒫 𝒫 𝑏 ∈ V |
11 | rabss 4050 | . . . . . . 7 ⊢ ({𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏 ↔ ∀𝑗 ∈ Top (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏)) | |
12 | pwuni 4877 | . . . . . . . . . 10 ⊢ 𝑗 ⊆ 𝒫 ∪ 𝑗 | |
13 | pweq 4557 | . . . . . . . . . 10 ⊢ (𝑏 = ∪ 𝑗 → 𝒫 𝑏 = 𝒫 ∪ 𝑗) | |
14 | 12, 13 | sseqtrrid 4022 | . . . . . . . . 9 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ⊆ 𝒫 𝑏) |
15 | velpw 4546 | . . . . . . . . 9 ⊢ (𝑗 ∈ 𝒫 𝒫 𝑏 ↔ 𝑗 ⊆ 𝒫 𝑏) | |
16 | 14, 15 | sylibr 236 | . . . . . . . 8 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏) |
17 | 16 | a1i 11 | . . . . . . 7 ⊢ (𝑗 ∈ Top → (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏)) |
18 | 11, 17 | mprgbir 3155 | . . . . . 6 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏 |
19 | 10, 18 | ssexi 5228 | . . . . 5 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ∈ V |
20 | 7, 8, 19 | fvmpt3i 6775 | . . . 4 ⊢ (𝐵 ∈ V → (TopOn‘𝐵) = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗}) |
21 | 20 | eleq2d 2900 | . . 3 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗})) |
22 | unieq 4851 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
23 | 22 | eqeq2d 2834 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝐵 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝐽)) |
24 | 23 | elrab 3682 | . . 3 ⊢ (𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗} ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)) |
25 | 21, 24 | syl6bb 289 | . 2 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))) |
26 | 1, 5, 25 | pm5.21nii 382 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 ⊆ wss 3938 𝒫 cpw 4541 ∪ cuni 4840 ‘cfv 6357 Topctop 21503 TopOnctopon 21520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-topon 21521 |
This theorem is referenced by: topontop 21523 toponuni 21524 toptopon 21527 toponcom 21538 istps2 21545 tgtopon 21581 distopon 21607 indistopon 21611 fctop 21614 cctop 21616 ppttop 21617 epttop 21619 mretopd 21702 toponmre 21703 resttopon 21771 resttopon2 21778 kgentopon 22148 txtopon 22201 pttopon 22206 xkotopon 22210 qtoptopon 22314 flimtopon 22580 fclstopon 22622 fclsfnflim 22637 utoptopon 22847 qtopt1 31101 neibastop1 33709 onsuctopon 33784 rfcnpre1 41283 cnfex 41292 icccncfext 42177 stoweidlem47 42339 |
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