Theorem bj-dmtopon 32242

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

Assertion

Ref	Expression
bj-dmtopon	⊢ dom TopOn = V

Proof of Theorem bj-dmtopon

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

Step	Hyp	Ref	Expression
1		vpwex 4775	. . . 4 ⊢ 𝒫 𝑥 ∈ V
2	1	pwex 4774	. . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V
3		eqcom 2617	. . . . . 6 ⊢ (𝑥 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝑥)
4	3	a1i 11	. . . . 5 ⊢ (𝑦 ∈ Top → (𝑥 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝑥))
5	4	rabbiia 3161	. . . 4 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥}
6		rabssab 3652	. . . . 5 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝑥}
7		bj-sspwpweq 32240	. . . . 5 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
8	6, 7	sstri 3577	. . . 4 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
9	5, 8	eqsstri 3598	. . 3 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝑥
10	2, 9	ssexi 4731	. 2 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ∈ V
11		df-topon 20523	. 2 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦})
12	10, 11	dmmpti 5936	1 ⊢ dom TopOn = V

Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173  𝒫 cpw 4108  ∪ cuni 4372  dom cdm 5038  Topctop 20517  TopOnctopon 20518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-fun 5806  df-fn 5807  df-topon 20523

This theorem is referenced by:  bj-fntopon  32243