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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 5377 | . . . 4 ⊢ 𝒫 𝑥 ∈ V | |
| 2 | 1 | pwex 5380 | . . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V |
| 3 | eqcom 2744 | . . . . 5 ⊢ (𝑥 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝑥) | |
| 4 | 3 | rabbii 3442 | . . . 4 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} |
| 5 | rabssab 4085 | . . . . 5 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝑥} | |
| 6 | pwpwssunieq 5104 | . . . . 5 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥 | |
| 7 | 5, 6 | sstri 3993 | . . . 4 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥 |
| 8 | 4, 7 | eqsstri 4030 | . . 3 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝑥 |
| 9 | 2, 8 | ssexi 5322 | . 2 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ∈ V |
| 10 | df-topon 22917 | . 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 1540 {cab 2714 {crab 3436 Vcvv 3480 𝒫 cpw 4600 ∪ cuni 4907 dom cdm 5685 Topctop 22899 TopOnctopon 22916 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-fun 6563 df-fn 6564 df-topon 22917 |
| This theorem is referenced by: fntopon 22930 |
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