Theorem fiinbas 23178

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 3956	. . . . . . . 8 ⊢ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)
2		eleq2 2851	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑥 ∩ 𝑦)))
3		sseq1 3959	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → (𝑤 ⊆ (𝑥 ∩ 𝑦) ↔ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦)))
4	2, 3	anbi12d 644	. . . . . . . . 9 ⊢ (𝑤 = (𝑥 ∩ 𝑦) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)) ↔ (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))))
5	4	rspcev 3579	. . . . . . . 8 ⊢ (((𝑥 ∩ 𝑦) ∈ 𝐵 ∧ (𝑧 ∈ (𝑥 ∩ 𝑦) ∧ (𝑥 ∩ 𝑦) ⊆ (𝑥 ∩ 𝑦))) → ∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
6	1, 5	mpanr2 717	. . . . . . 7 ⊢ (((𝑥 ∩ 𝑦) ∈ 𝐵 ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
7	6	ralrimiva 3156	. . . . . 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 3178	. . . 4 ⊢ (𝐵 ∈ 𝐶 → (∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵 → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
10	9	ralimdv 3178	. . 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 23174	. . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
12	10, 11	sylibrd 262	. 2 ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵 → 𝐵 ∈ TopBases))
13	12	imp 412	1 ⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088   ∩ cin 3901   ⊆ wss 3902  TopBasesctb 23171

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-in 3909  df-ss 3919  df-pw 4562  df-uni 4871  df-bases 23172

This theorem is used by:  fibas  23203  qtopbaslem  24985  ontopbas  37034  isbasisrelowl  38099