Theorem istopon 21086

Description: Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)

Assertion

Ref	Expression
istopon	⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))

Proof of Theorem istopon

Dummy variables 𝑏 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6466	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ V)
2		uniexg 7214	. . . 4 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
3		eleq1 2893	. . . 4 ⊢ (𝐵 = ∪ 𝐽 → (𝐵 ∈ V ↔ ∪ 𝐽 ∈ V))
4	2, 3	syl5ibrcom 239	. . 3 ⊢ (𝐽 ∈ Top → (𝐵 = ∪ 𝐽 → 𝐵 ∈ V))
5	4	imp 397	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽) → 𝐵 ∈ V)
6		eqeq1 2828	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝑗))
7	6	rabbidv 3401	. . . . 5 ⊢ (𝑏 = 𝐵 → {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗})
8		df-topon 21085	. . . . 5 ⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗})
9		vpwex 5076	. . . . . . 7 ⊢ 𝒫 𝑏 ∈ V
10	9	pwex 5079	. . . . . 6 ⊢ 𝒫 𝒫 𝑏 ∈ V
11		rabss 3903	. . . . . . 7 ⊢ ({𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏 ↔ ∀𝑗 ∈ Top (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏))
12		pwuni 4695	. . . . . . . . . 10 ⊢ 𝑗 ⊆ 𝒫 ∪ 𝑗
13		pweq 4380	. . . . . . . . . 10 ⊢ (𝑏 = ∪ 𝑗 → 𝒫 𝑏 = 𝒫 ∪ 𝑗)
14	12, 13	syl5sseqr 3878	. . . . . . . . 9 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ⊆ 𝒫 𝑏)
15		selpw 4384	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝒫 𝒫 𝑏 ↔ 𝑗 ⊆ 𝒫 𝑏)
16	14, 15	sylibr 226	. . . . . . . 8 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏)
17	16	a1i 11	. . . . . . 7 ⊢ (𝑗 ∈ Top → (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏))
18	11, 17	mprgbir 3135	. . . . . 6 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏
19	10, 18	ssexi 5027	. . . . 5 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ∈ V
20	7, 8, 19	fvmpt3i 6533	. . . 4 ⊢ (𝐵 ∈ V → (TopOn‘𝐵) = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗})
21	20	eleq2d 2891	. . 3 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗}))
22		unieq 4665	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
23	22	eqeq2d 2834	. . . 4 ⊢ (𝑗 = 𝐽 → (𝐵 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝐽))
24	23	elrab 3584	. . 3 ⊢ (𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗} ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))
25	21, 24	syl6bb 279	. 2 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)))
26	1, 5, 25	pm5.21nii 370	1 ⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  {crab 3120  Vcvv 3413   ⊆ wss 3797  𝒫 cpw 4377  ∪ cuni 4657  ‘cfv 6122  Topctop 21067  TopOnctopon 21084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-iota 6085  df-fun 6124  df-fv 6130  df-topon 21085

This theorem is referenced by:  topontop  21087  toponuni  21088  toptopon  21091  toponcom  21102  istps2  21109  tgtopon  21145  distopon  21171  indistopon  21175  fctop  21178  cctop  21180  ppttop  21181  epttop  21183  mretopd  21266  toponmre  21267  resttopon  21335  resttopon2  21342  kgentopon  21711  txtopon  21764  pttopon  21769  xkotopon  21773  qtoptopon  21877  flimtopon  22143  fclstopon  22185  fclsfnflim  22200  utoptopon  22409  qtopt1  30446  neibastop1  32891  onsuctopon  32965  rfcnpre1  39995  cnfex  40004  icccncfext  40894  stoweidlem47  41057