Theorem istopon 14804

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 14803	. . . . 5 ⊢ Fun TopOn
2		funrel 5350	. . . . 5 ⊢ (Fun TopOn → Rel TopOn)
3	1, 2	ax-mp 5	. . . 4 ⊢ Rel TopOn
4		relelfvdm 5680	. . . 4 ⊢ ((Rel TopOn ∧ 𝐽 ∈ (TopOn‘𝐵)) → 𝐵 ∈ dom TopOn)
5	3, 4	mpan 424	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ dom TopOn)
6	5	elexd 2817	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 ∈ V)
7		uniexg 4542	. . . 4 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
8		eleq1 2294	. . . 4 ⊢ (𝐵 = ∪ 𝐽 → (𝐵 ∈ V ↔ ∪ 𝐽 ∈ V))
9	7, 8	syl5ibrcom 157	. . 3 ⊢ (𝐽 ∈ Top → (𝐵 = ∪ 𝐽 → 𝐵 ∈ V))
10	9	imp 124	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽) → 𝐵 ∈ V)
11		eqeq1 2238	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝑗))
12	11	rabbidv 2792	. . . . 5 ⊢ (𝑏 = 𝐵 → {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗})
13		df-topon 14802	. . . . 5 ⊢ TopOn = (𝑏 ∈ V ↦ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗})
14		vpwex 4275	. . . . . . 7 ⊢ 𝒫 𝑏 ∈ V
15	14	pwex 4279	. . . . . 6 ⊢ 𝒫 𝒫 𝑏 ∈ V
16		rabss 3305	. . . . . . 7 ⊢ ({𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏 ↔ ∀𝑗 ∈ Top (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏))
17		pwuni 4288	. . . . . . . . . 10 ⊢ 𝑗 ⊆ 𝒫 ∪ 𝑗
18		pweq 3659	. . . . . . . . . 10 ⊢ (𝑏 = ∪ 𝑗 → 𝒫 𝑏 = 𝒫 ∪ 𝑗)
19	17, 18	sseqtrrid 3279	. . . . . . . . 9 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ⊆ 𝒫 𝑏)
20		velpw 3663	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝒫 𝒫 𝑏 ↔ 𝑗 ⊆ 𝒫 𝑏)
21	19, 20	sylibr 134	. . . . . . . 8 ⊢ (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏)
22	21	a1i 9	. . . . . . 7 ⊢ (𝑗 ∈ Top → (𝑏 = ∪ 𝑗 → 𝑗 ∈ 𝒫 𝒫 𝑏))
23	16, 22	mprgbir 2591	. . . . . 6 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ⊆ 𝒫 𝒫 𝑏
24	15, 23	ssexi 4232	. . . . 5 ⊢ {𝑗 ∈ Top ∣ 𝑏 = ∪ 𝑗} ∈ V
25	12, 13, 24	fvmpt3i 5735	. . . 4 ⊢ (𝐵 ∈ V → (TopOn‘𝐵) = {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗})
26	25	eleq2d 2301	. . 3 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ 𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗}))
27		unieq 3907	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
28	27	eqeq2d 2243	. . . 4 ⊢ (𝑗 = 𝐽 → (𝐵 = ∪ 𝑗 ↔ 𝐵 = ∪ 𝐽))
29	28	elrab 2963	. . 3 ⊢ (𝐽 ∈ {𝑗 ∈ Top ∣ 𝐵 = ∪ 𝑗} ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))
30	26, 29	bitrdi 196	. 2 ⊢ (𝐵 ∈ V → (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)))
31	6, 10, 30	pm5.21nii 712	1 ⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  {crab 2515  Vcvv 2803   ⊆ wss 3201  𝒫 cpw 3656  ∪ cuni 3898  dom cdm 4731  Rel wrel 4736  Fun wfun 5327  ‘cfv 5333  Topctop 14788  TopOnctopon 14801

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-topon 14802

This theorem is referenced by:  topontop  14805  toponuni  14806  toptopon  14809  toponcom  14818  istps2  14824  tgtopon  14857  distopon  14878  epttop  14881  resttopon  14962  resttopon2  14969  txtopon  15053