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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 3983 | . . . . . . 7 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ 𝐴 = ∪ 𝑦} | |
2 | eqcom 2801 | . . . . . . . 8 ⊢ (𝐴 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝐴) | |
3 | 2 | abbii 2860 | . . . . . . 7 ⊢ {𝑦 ∣ 𝐴 = ∪ 𝑦} = {𝑦 ∣ ∪ 𝑦 = 𝐴} |
4 | 1, 3 | sseqtri 3926 | . . . . . 6 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝐴} |
5 | pwpwssunieq 4927 | . . . . . 6 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴 | |
6 | 4, 5 | sstri 3900 | . . . . 5 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴 |
7 | pwexg 5173 | . . . . . 6 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
8 | 7 | pwexd 5174 | . . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V) |
9 | ssexg 5121 | . . . . 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 2798 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = ∪ 𝑦 ↔ 𝐴 = ∪ 𝑦)) | |
12 | 11 | rabbidv 3424 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦}) |
13 | df-topon 21203 | . . . . 5 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦}) | |
14 | 12, 13 | fvmptg 6636 | . . . 4 ⊢ ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V) → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦}) |
15 | 10, 14 | mpdan 683 | . . 3 ⊢ (𝐴 ∈ V → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦}) |
16 | 15, 6 | syl6eqss 3944 | . 2 ⊢ (𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴) |
17 | fvprc 6534 | . . 3 ⊢ (¬ 𝐴 ∈ V → (TopOn‘𝐴) = ∅) | |
18 | 0ss 4272 | . . 3 ⊢ ∅ ⊆ 𝒫 𝒫 𝐴 | |
19 | 17, 18 | syl6eqss 3944 | . 2 ⊢ (¬ 𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴) |
20 | 16, 19 | pm2.61i 183 | 1 ⊢ (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1522 ∈ wcel 2080 {cab 2774 {crab 3108 Vcvv 3436 ⊆ wss 3861 ∅c0 4213 𝒫 cpw 4455 ∪ cuni 4747 ‘cfv 6228 Topctop 21185 TopOnctopon 21202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ral 3109 df-rex 3110 df-rab 3113 df-v 3438 df-sbc 3708 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-op 4481 df-uni 4748 df-br 4965 df-opab 5027 df-mpt 5044 df-id 5351 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-iota 6192 df-fun 6230 df-fv 6236 df-topon 21203 |
This theorem is referenced by: toponmre 21385 |
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