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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 5281 | . . . 4 ⊢ 𝒫 𝑥 ∈ V | |
2 | 1 | pwex 5284 | . . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V |
3 | eqcom 2831 | . . . . 5 ⊢ (𝑥 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝑥) | |
4 | 3 | rabbii 3476 | . . . 4 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} |
5 | rabssab 4063 | . . . . 5 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝑥} | |
6 | pwpwssunieq 5029 | . . . . 5 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥 | |
7 | 5, 6 | sstri 3979 | . . . 4 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥 |
8 | 4, 7 | eqsstri 4004 | . . 3 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝑥 |
9 | 2, 8 | ssexi 5229 | . 2 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ∈ V |
10 | df-topon 21522 | . 2 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦}) | |
11 | 9, 10 | dmmpti 6495 | 1 ⊢ dom TopOn = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 {cab 2802 {crab 3145 Vcvv 3497 𝒫 cpw 4542 ∪ cuni 4841 dom cdm 5558 Topctop 21504 TopOnctopon 21521 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-fun 6360 df-fn 6361 df-topon 21522 |
This theorem is referenced by: fntopon 21535 |
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