Theorem eltg4i 20984

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 6361	. . . 4 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐵 ∈ dom topGen)
2		eltg 20981	. . . 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 256	. 2 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 ⊆ ∪ (𝐵 ∩ 𝒫 𝐴))
5		inss2 3980	. . . . 5 ⊢ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝒫 𝐴
6	5	unissi 4595	. . . 4 ⊢ ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ ∪ 𝒫 𝐴
7		unipw 5046	. . . 4 ⊢ ∪ 𝒫 𝐴 = 𝐴
8	6, 7	sseqtri 3784	. . 3 ⊢ ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴
9	8	a1i 11	. 2 ⊢ (𝐴 ∈ (topGen‘𝐵) → ∪ (𝐵 ∩ 𝒫 𝐴) ⊆ 𝐴)
10	4, 9	eqssd 3767	1 ⊢ (𝐴 ∈ (topGen‘𝐵) → 𝐴 = ∪ (𝐵 ∩ 𝒫 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1630   ∈ wcel 2144   ∩ cin 3720   ⊆ wss 3721  𝒫 cpw 4295  ∪ cuni 4572  dom cdm 5249  ‘cfv 6031  topGenctg 16305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-topgen 16311

This theorem is referenced by:  eltg3  20986  tgdom  21002  tgidm  21004  ontgval  32761