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Mirrors > Home > ILE Home > Th. List > dmtopon | GIF version |
Description: The domain of TopOn is V. (Contributed by BJ, 29-Apr-2021.) |
Ref | Expression |
---|---|
dmtopon | ⊢ dom TopOn = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vpwex 4152 | . . . 4 ⊢ 𝒫 𝑥 ∈ V | |
2 | 1 | pwex 4156 | . . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V |
3 | eqcom 2166 | . . . . 5 ⊢ (𝑥 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝑥) | |
4 | 3 | rabbii 2707 | . . . 4 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} |
5 | rabssab 3225 | . . . . 5 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝑥} | |
6 | pwpwssunieq 3948 | . . . . 5 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥 | |
7 | 5, 6 | sstri 3146 | . . . 4 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥 |
8 | 4, 7 | eqsstri 3169 | . . 3 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝑥 |
9 | 2, 8 | ssexi 4114 | . 2 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ∈ V |
10 | df-topon 12550 | . 2 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦}) | |
11 | 9, 10 | dmmpti 5311 | 1 ⊢ dom TopOn = V |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 {cab 2150 {crab 2446 Vcvv 2721 𝒫 cpw 3553 ∪ cuni 3783 dom cdm 4598 Topctop 12536 TopOnctopon 12549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-fun 5184 df-fn 5185 df-topon 12550 |
This theorem is referenced by: fntopon 12563 |
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