Theorem istopon 22868

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 6877	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ V)
2		uniexg 7695	. . . 4 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
3		eleq1 2825	. . . 4 ⊢ (𝐵 = ∪ 𝐽 → (𝐵 ∈ V ↔ ∪ 𝐽 ∈ V))
4	2, 3	syl5ibrcom 247	. . 3 ⊢ (𝐽 ∈ Top → (𝐵 = ∪ 𝐽 → 𝐵 ∈ V))
5	4	imp 406	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽) → 𝐵 ∈ V)
6		eqeq1 2741	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝑗))
7	6	rabbidv 3408	. . . . 5 ⊢ (𝑏 = 𝐵 → {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗})
8		df-topon 22867	. . . . 5 ⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗})
9		vpwex 5324	. . . . . . 7 ⊢ 𝒫 𝑏 ∈ V
10	9	pwex 5327	. . . . . 6 ⊢ 𝒫 𝒫 𝑏 ∈ V
11		rabss 4024	. . . . . . 7 ⊢ ({𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏 ↔ ∀𝑗 ∈ Top (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏))
12		pwuni 4903	. . . . . . . . . 10 ⊢ 𝑗 ⊆ 𝒫 ∪ 𝑗
13		pweq 4570	. . . . . . . . . 10 ⊢ (𝑏 = ∪ 𝑗 → 𝒫 𝑏 = 𝒫 ∪ 𝑗)
14	12, 13	sseqtrrid 3979	. . . . . . . . 9 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ⊆ 𝒫 𝑏)
15		velpw 4561	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝒫 𝒫 𝑏 ↔ 𝑗 ⊆ 𝒫 𝑏)
16	14, 15	sylibr 234	. . . . . . . 8 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏)
17	16	a1i 11	. . . . . . 7 ⊢ (𝑗 ∈ Top → (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏))
18	11, 17	mprgbir 3059	. . . . . 6 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏
19	10, 18	ssexi 5269	. . . . 5 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ∈ V
20	7, 8, 19	fvmpt3i 6955	. . . 4 ⊢ (𝐵 ∈ V → (TopOn‘𝐵) = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗})
21	20	eleq2d 2823	. . 3 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗}))
22		unieq 4876	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
23	22	eqeq2d 2748	. . . 4 ⊢ (𝑗 = 𝐽 → (𝐵 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝐽))
24	23	elrab 3648	. . 3 ⊢ (𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗} ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))
25	21, 24	bitrdi 287	. 2 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)))
26	1, 5, 25	pm5.21nii 378	1 ⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3401  Vcvv 3442   ⊆ wss 3903  𝒫 cpw 4556  ∪ cuni 4865  ‘cfv 6500  Topctop 22849  TopOnctopon 22866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-topon 22867

This theorem is referenced by:  topontop  22869  toponuni  22870  toptopon  22873  toponcom  22884  istps2  22891  tgtopon  22927  distopon  22953  indistopon  22957  fctop  22960  cctop  22962  ppttop  22963  epttop  22965  mretopd  23048  toponmre  23049  resttopon  23117  resttopon2  23124  kgentopon  23494  txtopon  23547  pttopon  23552  xkotopon  23556  qtoptopon  23660  flimtopon  23926  fclstopon  23968  fclsfnflim  23983  utoptopon  24192  qtopt1  34012  neibastop1  36572  onsuctopon  36647  rfcnpre1  45373  cnfex  45382  icccncfext  46239  stoweidlem47  46399