Theorem eltg4i 22018

Description: An open set in a topology generated by a basis is the union of all basic open sets contained in it. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
eltg4i	⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 = ∪ (𝐵 ∩ 𝒫 𝐴))

Proof of Theorem eltg4i

Step	Hyp	Ref	Expression
1		elfvdm 6788	. . . 4 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen)
2		eltg 22015	. . . 4 ⊢ (𝐵 ∈ dom topGen → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ (topGen‘𝐵) → (𝐴 ∈ (topGen‘𝐵) ↔ 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴)))
4	3	ibi 266	. 2 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴))
5		inss2 4160	. . . . 5 ⊢ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
6	5	unissi 4845	. . . 4 ⊢ ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ ∪ 𝒫 𝐴
7		unipw 5360	. . . 4 ⊢ ∪ 𝒫 𝐴 = 𝐴
8	6, 7	sseqtri 3953	. . 3 ⊢ ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴
9	8	a1i 11	. 2 ⊢ (𝐴 ∈ (topGen‘𝐵) → ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴)
10	4, 9	eqssd 3934	1 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 = ∪ (𝐵 ∩ 𝒫 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836  dom cdm 5580  ‘cfv 6418  topGenctg 17065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-topgen 17071

This theorem is referenced by:  eltg3  22020  tgdom  22036  tgidm  22038  ontgval  34547