Theorem dmtopon 20727

Description: The domain of TopOn is 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 4849	. . . 4 ⊢ 𝒫 𝑥 ∈ V
2	1	pwex 4848	. . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V
3		eqcom 2629	. . . . . 6 ⊢ (𝑥 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝑥)
4	3	a1i 11	. . . . 5 ⊢ (𝑦 ∈ Top → (𝑥 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝑥))
5	4	rabbiia 3185	. . . 4 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥}
6		rabssab 3690	. . . . 5 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝑥}
7		pwpwssunieq 4615	. . . . 5 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
8	6, 7	sstri 3612	. . . 4 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
9	5, 8	eqsstri 3635	. . 3 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝑥
10	2, 9	ssexi 4803	. 2 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ∈ V
11		df-topon 20716	. 2 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦})
12	10, 11	dmmpti 6023	1 ⊢ dom TopOn = V

Syntax hints:   ↔ wb 196   = wceq 1483   ∈ wcel 1990  {cab 2608  {crab 2916  Vcvv 3200  𝒫 cpw 4158  ∪ cuni 4436  dom cdm 5114  Topctop 20698  TopOnctopon 20715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-fun 5890  df-fn 5891  df-topon 20716

This theorem is referenced by:  fntopon  20728