Theorem dmtopon 23132

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 5350	. . . 4 ⊢ 𝒫 𝑥 ∈ V
2	1	pwex 5353	. . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V
3		eqcom 2772	. . . . 5 ⊢ (𝑥 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝑥)
4	3	rabbii 3423	. . . 4 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥}
5		rabssab 4040	. . . . 5 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝑥}
6		pwpwssunieq 5072	. . . . 5 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
7	5, 6	sstri 3947	. . . 4 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
8	4, 7	eqsstri 3984	. . 3 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝑥
9	2, 8	ssexi 5295	. 2 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ∈ V
10		df-topon 23120	. 2 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦})
11	9, 10	dmmpti 6683	1 ⊢ dom TopOn = V

This proof depends on syntax axioms:   = wceq 1570  {cab 2743  {crab 3418  Vcvv 3457  𝒫 cpw 4564  ∪ cuni 4874  dom cdm 5663  Topctop 23102  TopOnctopon 23119

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-fun 6542  df-fn 6543  df-topon 23120

This theorem is used by:  fntopon  23133