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Mirrors > Home > MPE Home > Th. List > dmtopon | Structured version Visualization version GIF version |
Description: The domain of TopOn is the universal class V. (Contributed by BJ, 29-Apr-2021.) |
Ref | Expression |
---|---|
dmtopon | ⊢ dom TopOn = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vpwex 5336 | . . . 4 ⊢ 𝒫 𝑥 ∈ V | |
2 | 1 | pwex 5339 | . . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V |
3 | eqcom 2740 | . . . . 5 ⊢ (𝑥 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝑥) | |
4 | 3 | rabbii 3412 | . . . 4 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} |
5 | rabssab 4047 | . . . . 5 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝑥} | |
6 | pwpwssunieq 5068 | . . . . 5 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥 | |
7 | 5, 6 | sstri 3957 | . . . 4 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥 |
8 | 4, 7 | eqsstri 3982 | . . 3 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝑥 |
9 | 2, 8 | ssexi 5283 | . 2 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ∈ V |
10 | df-topon 22283 | . 2 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦}) | |
11 | 9, 10 | dmmpti 6649 | 1 ⊢ dom TopOn = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 {cab 2710 {crab 3406 Vcvv 3447 𝒫 cpw 4564 ∪ cuni 4869 dom cdm 5637 Topctop 22265 TopOnctopon 22282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-fun 6502 df-fn 6503 df-topon 22283 |
This theorem is referenced by: fntopon 22296 |
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