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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 5277 | . . . 4 ⊢ 𝒫 𝑥 ∈ V | |
2 | 1 | pwex 5280 | . . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V |
3 | eqcom 2828 | . . . . 5 ⊢ (𝑥 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝑥) | |
4 | 3 | rabbii 3473 | . . . 4 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} |
5 | rabssab 4059 | . . . . 5 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝑥} | |
6 | pwpwssunieq 5025 | . . . . 5 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥 | |
7 | 5, 6 | sstri 3975 | . . . 4 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥 |
8 | 4, 7 | eqsstri 4000 | . . 3 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝑥 |
9 | 2, 8 | ssexi 5225 | . 2 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ∈ V |
10 | df-topon 21518 | . 2 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦}) | |
11 | 9, 10 | dmmpti 6491 | 1 ⊢ dom TopOn = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 {cab 2799 {crab 3142 Vcvv 3494 𝒫 cpw 4538 ∪ cuni 4837 dom cdm 5554 Topctop 21500 TopOnctopon 21517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-fun 6356 df-fn 6357 df-topon 21518 |
This theorem is referenced by: fntopon 21531 |
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