Theorem fiinbas 22898

Description: If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
fiinbas	⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem fiinbas

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3955	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)
2		eleq2 2824	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑥 ∩ 𝑦)))
3		sseq1 3958	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → (𝑤 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
4	2, 3	anbi12d 633	. . . . . . . . 9 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)) ↔ (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))))
5	4	rspcev 3575	. . . . . . . 8 ⊢ (((𝑥 ∩ 𝑦) ∈ 𝐵 ∧ (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))) → ∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
6	1, 5	mpanr2 705	. . . . . . 7 ⊢ (((𝑥 ∩ 𝑦) ∈ 𝐵 ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
7	6	ralrimiva 3127	. . . . . 6 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝐵 → ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
8	7	a1i 11	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → ((𝑥 ∩ 𝑦) ∈ 𝐵 → ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
9	8	ralimdv 3149	. . . 4 ⊢ (𝐵 ∈ 𝐶 → (∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
10	9	ralimdv 3149	. . 3 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
11		isbasis2g 22894	. . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
12	10, 11	sylibrd 259	. 2 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵 → 𝐵 ∈ TopBases))
13	12	imp 406	1 ⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3050  ∃wrex 3059   ∩ cin 3899   ⊆ wss 3900  TopBasesctb 22891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-in 3907  df-ss 3917  df-pw 4555  df-uni 4863  df-bases 22892

This theorem is referenced by:  fibas  22923  qtopbaslem  24704  ontopbas  36601  isbasisrelowl  37532