| 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 4085 | . . . . . . 7 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ 𝐴 = ∪ 𝑦} | |
| 2 | eqcom 2744 | . . . . . . . 8 ⊢ (𝐴 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝐴) | |
| 3 | 2 | abbii 2809 | . . . . . . 7 ⊢ {𝑦 ∣ 𝐴 = ∪ 𝑦} = {𝑦 ∣ ∪ 𝑦 = 𝐴} |
| 4 | 1, 3 | sseqtri 4032 | . . . . . 6 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝐴} |
| 5 | pwpwssunieq 5104 | . . . . . 6 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴 | |
| 6 | 4, 5 | sstri 3993 | . . . . 5 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴 |
| 7 | pwexg 5378 | . . . . . 6 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
| 8 | 7 | pwexd 5379 | . . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V) |
| 9 | ssexg 5323 | . . . . 5 ⊢ (({𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴 ∈ V) → {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V) | |
| 10 | 6, 8, 9 | sylancr 587 | . . . 4 ⊢ (𝐴 ∈ V → {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V) |
| 11 | eqeq1 2741 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = ∪ 𝑦 ↔ 𝐴 = ∪ 𝑦)) | |
| 12 | 11 | rabbidv 3444 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦}) |
| 13 | df-topon 22917 | . . . . 5 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦}) | |
| 14 | 12, 13 | fvmptg 7014 | . . . 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 4028 | . 2 ⊢ (𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴) |
| 17 | fvprc 6898 | . . 3 ⊢ (¬ 𝐴 ∈ V → (TopOn‘𝐴) = ∅) | |
| 18 | 0ss 4400 | . . 3 ⊢ ∅ ⊆ 𝒫 𝒫 𝐴 | |
| 19 | 17, 18 | eqsstrdi 4028 | . 2 ⊢ (¬ 𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴) |
| 20 | 16, 19 | pm2.61i 182 | 1 ⊢ (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 {cab 2714 {crab 3436 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 ∪ cuni 4907 ‘cfv 6561 Topctop 22899 TopOnctopon 22916 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-topon 22917 |
| This theorem is referenced by: toponmre 23101 |
| Copyright terms: Public domain | W3C validator |