Theorem eltg4i 22903

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 6918	. . . 4 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen)
2		eltg 22900	. . . 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 267	. 2 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴))
5		inss2 4218	. . . . 5 ⊢ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
6	5	unissi 4897	. . . 4 ⊢ ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ ∪ 𝒫 𝐴
7		unipw 5430	. . . 4 ⊢ ∪ 𝒫 𝐴 = 𝐴
8	6, 7	sseqtri 4012	. . 3 ⊢ ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴
9	8	a1i 11	. 2 ⊢ (𝐴 ∈ (topGen‘𝐵) → ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴)
10	4, 9	eqssd 3981	1 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 = ∪ (𝐵 ∩ 𝒫 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109   ∩ cin 3930   ⊆ wss 3931  𝒫 cpw 4580  ∪ cuni 4888  dom cdm 5659  ‘cfv 6536  topGenctg 17456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-topgen 17462

This theorem is referenced by:  eltg3  22905  tgdom  22921  tgidm  22923  ontgval  36454