Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > toponsspwpw | Structured version Visualization version GIF version |
Description: The set of topologies on a set is included in the double power set of that set. (Contributed by BJ, 29-Apr-2021.) |
Ref | Expression |
---|---|
toponsspwpw | ⊢ (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabssab 3998 | . . . . . . 7 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ 𝐴 = ∪ 𝑦} | |
2 | eqcom 2744 | . . . . . . . 8 ⊢ (𝐴 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝐴) | |
3 | 2 | abbii 2808 | . . . . . . 7 ⊢ {𝑦 ∣ 𝐴 = ∪ 𝑦} = {𝑦 ∣ ∪ 𝑦 = 𝐴} |
4 | 1, 3 | sseqtri 3937 | . . . . . 6 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝐴} |
5 | pwpwssunieq 5012 | . . . . . 6 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴 | |
6 | 4, 5 | sstri 3910 | . . . . 5 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴 |
7 | pwexg 5271 | . . . . . 6 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
8 | 7 | pwexd 5272 | . . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V) |
9 | ssexg 5216 | . . . . 5 ⊢ (({𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴 ∈ V) → {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V) | |
10 | 6, 8, 9 | sylancr 590 | . . . 4 ⊢ (𝐴 ∈ V → {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V) |
11 | eqeq1 2741 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = ∪ 𝑦 ↔ 𝐴 = ∪ 𝑦)) | |
12 | 11 | rabbidv 3390 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦}) |
13 | df-topon 21808 | . . . . 5 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦}) | |
14 | 12, 13 | fvmptg 6816 | . . . 4 ⊢ ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V) → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦}) |
15 | 10, 14 | mpdan 687 | . . 3 ⊢ (𝐴 ∈ V → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦}) |
16 | 15, 6 | eqsstrdi 3955 | . 2 ⊢ (𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴) |
17 | fvprc 6709 | . . 3 ⊢ (¬ 𝐴 ∈ V → (TopOn‘𝐴) = ∅) | |
18 | 0ss 4311 | . . 3 ⊢ ∅ ⊆ 𝒫 𝒫 𝐴 | |
19 | 17, 18 | eqsstrdi 3955 | . 2 ⊢ (¬ 𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴) |
20 | 16, 19 | pm2.61i 185 | 1 ⊢ (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1543 ∈ wcel 2110 {cab 2714 {crab 3065 Vcvv 3408 ⊆ wss 3866 ∅c0 4237 𝒫 cpw 4513 ∪ cuni 4819 ‘cfv 6380 Topctop 21790 TopOnctopon 21807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-topon 21808 |
This theorem is referenced by: toponmre 21990 |
Copyright terms: Public domain | W3C validator |