Theorem fbasssin 24030

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 6922	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
2		isfbas2 24029	. . . . . . 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 4169	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ∩ 𝑧) = (𝐴 ∩ 𝑧))
8	7	sseq2d 3972	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ⊆ (𝑦 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐴 ∩ 𝑧)))
9	8	rexbidv 3192	. . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧) ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝑧)))
10		ineq2 4170	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝐴 ∩ 𝑧) = (𝐴 ∩ 𝐵))
11	10	sseq2d 3972	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑥 ⊆ (𝐴 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)))
12	11	rexbidv 3192	. . . 4 ⊢ (𝑧 = 𝐵 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝑧) ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵)))
13	9, 12	rspc2v 3595	. . 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 2146   ≠ wne 2961   ∉ wnel 3067  ∀wral 3082  ∃wrex 3092   ∩ cin 3907   ⊆ wss 3908  ∅c0 4289  𝒫 cpw 4567  dom cdm 5666  ‘cfv 6543  fBascfbas 21547

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fv 6551  df-fbas 21556

This theorem is used by:  fbssfi  24031  fbncp  24033  fbun  24034  fbfinnfr  24035  trfbas2  24037  filin  24048  fgcl  24072  fbasrn  24078