Theorem fbasssin 23774

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 6913	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
2		isfbas2 23773	. . . . . . 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 17	. . . . . 6 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐹 ∈ (fBas‘𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧)))))
4	3	ibi 267	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧))))
5	4	simprd 495	. . . 4 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧)))
6	5	simp3d 1144	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧))
7		ineq1 4188	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ∩ 𝑧) = (𝐴 ∩ 𝑧))
8	7	sseq2d 3991	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ⊆ (𝑦 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐴 ∩ 𝑧)))
9	8	rexbidv 3164	. . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧) ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝑧)))
10		ineq2 4189	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝐴 ∩ 𝑧) = (𝐴 ∩ 𝐵))
11	10	sseq2d 3991	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑥 ⊆ (𝐴 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)))
12	11	rexbidv 3164	. . . 4 ⊢ (𝑧 = 𝐵 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝑧) ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵)))
13	9, 12	rspc2v 3612	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵)))
14	6, 13	syl5com 31	. 2 ⊢ (𝐹 ∈ (fBas‘𝑋) → ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵)))
15	14	3impib 1116	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932   ∉ wnel 3036  ∀wral 3051  ∃wrex 3060   ∩ cin 3925   ⊆ wss 3926  ∅c0 4308  𝒫 cpw 4575  dom cdm 5654  ‘cfv 6531  fBascfbas 21303

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fv 6539  df-fbas 21312

This theorem is referenced by:  fbssfi  23775  fbncp  23777  fbun  23778  fbfinnfr  23779  trfbas2  23781  filin  23792  fgcl  23816  fbasrn  23822