Theorem fbasssin 24188

Description: A filter base contains subsets of its pairwise intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Jeff Hankins, 1-Dec-2010.)

Assertion

Ref	Expression
fbasssin	⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝑋

Proof of Theorem fbasssin

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

Step	Hyp	Ref	Expression
1		elfvdm 6945	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
2		isfbas2 24187	. . . . . . 7 ⊢ (𝑋 ∈ dom fBas → (𝐹 ∈ (fBas‘𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧)))))
3	1, 2	syl 18	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐹 ∈ (fBas‘𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧)))))
4	3	ibi 270	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧))))
5	4	simprd 501	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧)))
6	5	simp3d 1162	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧))
7		ineq1 4113	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ∩ 𝑧) = (𝐴 ∩ 𝑧))
8	7	sseq2d 3945	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ⊆ (𝑦 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐴 ∩ 𝑧)))
9	8	rexbidv 3190	. . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧) ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝑧)))
10		ineq2 4114	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝐴 ∩ 𝑧) = (𝐴 ∩ 𝐵))
11	10	sseq2d 3945	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑥 ⊆ (𝐴 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)))
12	11	rexbidv 3190	. . . 4 ⊢ (𝑧 = 𝐵 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝑧) ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵)))
13	9, 12	rspc2v 3590	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵)))
14	6, 13	syl5com 32	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵)))
15	14	3impib 1134	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ∉ wnel 3065  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  𝒫 cpw 4560  dom cdm 5656  ‘cfv 6561  fBascfbas 21699

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fv 6569  df-fbas 21708

This theorem is used by:  fbssfi  24189  fbncp  24191  fbun  24192  fbfinnfr  24193  trfbas2  24195  filin  24206  fgcl  24230  fbasrn  24236