Theorem dmtopon 14570

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 4231	. . . 4 ⊢ 𝒫 𝑥 ∈ V
2	1	pwex 4235	. . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V
3		eqcom 2208	. . . . 5 ⊢ (𝑥 = ∪ 𝑦 ↔ ∪ 𝑦 = 𝑥)
4	3	rabbii 2759	. . . 4 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} = {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥}
5		rabssab 3285	. . . . 5 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ {𝑦 ∣ ∪ 𝑦 = 𝑥}
6		pwpwssunieq 4022	. . . . 5 ⊢ {𝑦 ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
7	5, 6	sstri 3206	. . . 4 ⊢ {𝑦 ∈ Top ∣ ∪ 𝑦 = 𝑥} ⊆ 𝒫 𝒫 𝑥
8	4, 7	eqsstri 3229	. . 3 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ⊆ 𝒫 𝒫 𝑥
9	2, 8	ssexi 4190	. 2 ⊢ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦} ∈ V
10		df-topon 14558	. 2 ⊢ TopOn = (𝑥 ∈ V ↦ {𝑦 ∈ Top ∣ 𝑥 = ∪ 𝑦})
11	9, 10	dmmpti 5415	1 ⊢ dom TopOn = V

Syntax hints:   = wceq 1373  {cab 2192  {crab 2489  Vcvv 2773  𝒫 cpw 3621  ∪ cuni 3856  dom cdm 4683  Topctop 14544  TopOnctopon 14557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-fun 5282  df-fn 5283  df-topon 14558

This theorem is referenced by:  fntopon  14571