Theorem istopon 22902

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 6869	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ V)
2		uniexg 7690	. . . 4 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
3		eleq1 2828	. . . 4 ⊢ (𝐵 = ∪ 𝐽 → (𝐵 ∈ V ↔ ∪ 𝐽 ∈ V))
4	2, 3	syl5ibrcom 248	. . 3 ⊢ (𝐽 ∈ Top → (𝐵 = ∪ 𝐽 → 𝐵 ∈ V))
5	4	imp 407	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽) → 𝐵 ∈ V)
6		eqeq1 2744	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝑗))
7	6	rabbidv 3399	. . . . 5 ⊢ (𝑏 = 𝐵 → {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗})
8		df-topon 22901	. . . . 5 ⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗})
9		vpwex 5313	. . . . . . 7 ⊢ 𝒫 𝑏 ∈ V
10	9	pwex 5316	. . . . . 6 ⊢ 𝒫 𝒫 𝑏 ∈ V
11		rabss 4008	. . . . . . 7 ⊢ ({𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏 ↔ ∀𝑗 ∈ Top (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏))
12		pwuni 4883	. . . . . . . . . 10 ⊢ 𝑗 ⊆ 𝒫 ∪ 𝑗
13		pweq 4550	. . . . . . . . . 10 ⊢ (𝑏 = ∪ 𝑗 → 𝒫 𝑏 = 𝒫 ∪ 𝑗)
14	12, 13	sseqtrrid 3965	. . . . . . . . 9 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ⊆ 𝒫 𝑏)
15		velpw 4541	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝒫 𝒫 𝑏 ↔ 𝑗 ⊆ 𝒫 𝑏)
16	14, 15	sylibr 235	. . . . . . . 8 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏)
17	16	a1i 11	. . . . . . 7 ⊢ (𝑗 ∈ Top → (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏))
18	11, 17	mprgbir 3061	. . . . . 6 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏
19	10, 18	ssexi 5257	. . . . 5 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ∈ V
20	7, 8, 19	fvmpt3i 6948	. . . 4 ⊢ (𝐵 ∈ V → (TopOn‘𝐵) = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗})
21	20	eleq2d 2826	. . 3 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗}))
22		unieq 4856	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
23	22	eqeq2d 2751	. . . 4 ⊢ (𝑗 = 𝐽 → (𝐵 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝐽))
24	23	elrab 3636	. . 3 ⊢ (𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗} ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))
25	21, 24	bitrdi 288	. 2 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)))
26	1, 5, 25	pm5.21nii 379	1 ⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3392  Vcvv 3432   ⊆ wss 3890  𝒫 cpw 4536  ∪ cuni 4845  ‘cfv 6492  Topctop 22883  TopOnctopon 22900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-topon 22901

This theorem is referenced by:  topontop  22903  toponuni  22904  toptopon  22907  toponcom  22918  istps2  22925  tgtopon  22961  distopon  22987  indistopon  22991  fctop  22994  cctop  22996  ppttop  22997  epttop  22999  mretopd  23082  toponmre  23083  resttopon  23151  resttopon2  23158  kgentopon  23528  txtopon  23581  pttopon  23586  xkotopon  23590  qtoptopon  23694  flimtopon  23960  fclstopon  24002  fclsfnflim  24017  utoptopon  24226  qtopt1  34026  neibastop1  36594  onsuctopon  36669  rfcnpre1  45474  cnfex  45483  icccncfext  46337  stoweidlem47  46497