Theorem dmtopon 22817

Description: The domain of TopOn is the universal class V. (Contributed by BJ, 29-Apr-2021.)

Assertion

Ref	Expression
dmtopon	⊢ dom TopOn = V

Proof of Theorem dmtopon

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vpwex 5335	. . . 4 ⊢ 𝒫 𝑥 ∈ V
2	1	pwex 5338	. . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V
3		eqcom 2737	. . . . 5 ⊢ (𝑥 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝑥)
4	3	rabbii 3414	. . . 4 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥}
5		rabssab 4051	. . . . 5 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝑥}
6		pwpwssunieq 5071	. . . . 5 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
7	5, 6	sstri 3959	. . . 4 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
8	4, 7	eqsstri 3996	. . 3 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝑥
9	2, 8	ssexi 5280	. 2 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ∈ V
10		df-topon 22805	. 2 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦})
11	9, 10	dmmpti 6665	1 ⊢ dom TopOn = V

Syntax hints:   = wceq 1540  {cab 2708  {crab 3408  Vcvv 3450  𝒫 cpw 4566  ∪ cuni 4874  dom cdm 5641  Topctop 22787  TopOnctopon 22804

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-fun 6516  df-fn 6517  df-topon 22805

This theorem is referenced by:  fntopon  22818