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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 5382 | . . . 4 ⊢ 𝒫 𝑥 ∈ V | |
2 | 1 | pwex 5385 | . . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V |
3 | eqcom 2741 | . . . . 5 ⊢ (𝑥 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝑥) | |
4 | 3 | rabbii 3438 | . . . 4 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} |
5 | rabssab 4094 | . . . . 5 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝑥} | |
6 | pwpwssunieq 5108 | . . . . 5 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥 | |
7 | 5, 6 | sstri 4004 | . . . 4 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥 |
8 | 4, 7 | eqsstri 4029 | . . 3 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝑥 |
9 | 2, 8 | ssexi 5327 | . 2 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ∈ V |
10 | df-topon 22932 | . 2 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦}) | |
11 | 9, 10 | dmmpti 6712 | 1 ⊢ dom TopOn = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 {cab 2711 {crab 3432 Vcvv 3477 𝒫 cpw 4604 ∪ cuni 4911 dom cdm 5688 Topctop 22914 TopOnctopon 22931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-fun 6564 df-fn 6565 df-topon 22932 |
This theorem is referenced by: fntopon 22945 |
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