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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 3690 | . . . . . . 7 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ 𝐴 = ∪ 𝑦} | |
2 | eqcom 2629 | . . . . . . . 8 ⊢ (𝐴 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝐴) | |
3 | 2 | abbii 2739 | . . . . . . 7 ⊢ {𝑦 ∣ 𝐴 = ∪ 𝑦} = {𝑦 ∣ ∪ 𝑦 = 𝐴} |
4 | 1, 3 | sseqtri 3637 | . . . . . 6 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝐴} |
5 | pwpwssunieq 4615 | . . . . . 6 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝐴} ⊆ 𝒫 𝒫 𝐴 | |
6 | 4, 5 | sstri 3612 | . . . . 5 ⊢ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴 |
7 | pwexg 4850 | . . . . . 6 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
8 | pwexg 4850 | . . . . . 6 ⊢ (𝒫 𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝒫 𝐴 ∈ V) |
10 | ssexg 4804 | . . . . 5 ⊢ (({𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝐴 ∧ 𝒫 𝒫 𝐴 ∈ V) → {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V) | |
11 | 6, 9, 10 | sylancr 695 | . . . 4 ⊢ (𝐴 ∈ V → {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V) |
12 | eqeq1 2626 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = ∪ 𝑦 ↔ 𝐴 = ∪ 𝑦)) | |
13 | 12 | rabbidv 3189 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦}) |
14 | df-topon 20716 | . . . . 5 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦}) | |
15 | 13, 14 | fvmptg 6280 | . . . 4 ⊢ ((𝐴 ∈ V ∧ {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦} ∈ V) → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦}) |
16 | 11, 15 | mpdan 702 | . . 3 ⊢ (𝐴 ∈ V → (TopOn‘𝐴) = {𝑦 ∈ Top ∣ 𝐴 = ∪ 𝑦}) |
17 | 16, 6 | syl6eqss 3655 | . 2 ⊢ (𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴) |
18 | fvprc 6185 | . . 3 ⊢ (¬ 𝐴 ∈ V → (TopOn‘𝐴) = ∅) | |
19 | 0ss 3972 | . . 3 ⊢ ∅ ⊆ 𝒫 𝒫 𝐴 | |
20 | 18, 19 | syl6eqss 3655 | . 2 ⊢ (¬ 𝐴 ∈ V → (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴) |
21 | 17, 20 | pm2.61i 176 | 1 ⊢ (TopOn‘𝐴) ⊆ 𝒫 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1483 ∈ wcel 1990 {cab 2608 {crab 2916 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 ∪ cuni 4436 ‘cfv 5888 Topctop 20698 TopOnctopon 20715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-topon 20716 |
This theorem is referenced by: (None) |
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