Theorem istopon 12805

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		funtopon 12804	. . . . 5 ⊢ Fun TopOn
2		funrel 5215	. . . . 5 ⊢ (Fun TopOn → Rel TopOn)
3	1, 2	ax-mp 5	. . . 4 ⊢ Rel TopOn
4		relelfvdm 5528	. . . 4 ⊢ ((Rel TopOn ∧ 𝐽 ∈ (TopOn‘𝐵)) → 𝐵 ∈ dom TopOn)
5	3, 4	mpan 422	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ dom TopOn)
6	5	elexd 2743	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ V)
7		uniexg 4424	. . . 4 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
8		eleq1 2233	. . . 4 ⊢ (𝐵 = ∪ 𝐽 → (𝐵 ∈ V ↔ ∪ 𝐽 ∈ V))
9	7, 8	syl5ibrcom 156	. . 3 ⊢ (𝐽 ∈ Top → (𝐵 = ∪ 𝐽 → 𝐵 ∈ V))
10	9	imp 123	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽) → 𝐵 ∈ V)
11		eqeq1 2177	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝑗))
12	11	rabbidv 2719	. . . . 5 ⊢ (𝑏 = 𝐵 → {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗})
13		df-topon 12803	. . . . 5 ⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗})
14		vpwex 4165	. . . . . . 7 ⊢ 𝒫 𝑏 ∈ V
15	14	pwex 4169	. . . . . 6 ⊢ 𝒫 𝒫 𝑏 ∈ V
16		rabss 3224	. . . . . . 7 ⊢ ({𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏 ↔ ∀𝑗 ∈ Top (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏))
17		pwuni 4178	. . . . . . . . . 10 ⊢ 𝑗 ⊆ 𝒫 ∪ 𝑗
18		pweq 3569	. . . . . . . . . 10 ⊢ (𝑏 = ∪ 𝑗 → 𝒫 𝑏 = 𝒫 ∪ 𝑗)
19	17, 18	sseqtrrid 3198	. . . . . . . . 9 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ⊆ 𝒫 𝑏)
20		velpw 3573	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝒫 𝒫 𝑏 ↔ 𝑗 ⊆ 𝒫 𝑏)
21	19, 20	sylibr 133	. . . . . . . 8 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏)
22	21	a1i 9	. . . . . . 7 ⊢ (𝑗 ∈ Top → (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏))
23	16, 22	mprgbir 2528	. . . . . 6 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏
24	15, 23	ssexi 4127	. . . . 5 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ∈ V
25	12, 13, 24	fvmpt3i 5576	. . . 4 ⊢ (𝐵 ∈ V → (TopOn‘𝐵) = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗})
26	25	eleq2d 2240	. . 3 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗}))
27		unieq 3805	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
28	27	eqeq2d 2182	. . . 4 ⊢ (𝑗 = 𝐽 → (𝐵 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝐽))
29	28	elrab 2886	. . 3 ⊢ (𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗} ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))
30	26, 29	bitrdi 195	. 2 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)))
31	6, 10, 30	pm5.21nii 699	1 ⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141  {crab 2452  Vcvv 2730   ⊆ wss 3121  𝒫 cpw 3566  ∪ cuni 3796  dom cdm 4611  Rel wrel 4616  Fun wfun 5192  ‘cfv 5198  Topctop 12789  TopOnctopon 12802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-topon 12803

This theorem is referenced by:  topontop  12806  toponuni  12807  toptopon  12810  toponcom  12819  istps2  12825  tgtopon  12860  distopon  12881  epttop  12884  resttopon  12965  resttopon2  12972  txtopon  13056