Theorem fbasssin 23811

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 6868	. . . . . . 7 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
2		isfbas2 23810	. . . . . . 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 1145	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → ∀𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝐹 ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧))
7		ineq1 4154	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ∩ 𝑧) = (𝐴 ∩ 𝑧))
8	7	sseq2d 3955	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ⊆ (𝑦 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐴 ∩ 𝑧)))
9	8	rexbidv 3162	. . . 4 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝑦 ∩ 𝑧) ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝑧)))
10		ineq2 4155	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝐴 ∩ 𝑧) = (𝐴 ∩ 𝐵))
11	10	sseq2d 3955	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑥 ⊆ (𝐴 ∩ 𝑧) ↔ 𝑥 ⊆ (𝐴 ∩ 𝐵)))
12	11	rexbidv 3162	. . . 4 ⊢ (𝑧 = 𝐵 → (∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝑧) ↔ ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵)))
13	9, 12	rspc2v 3576	. . 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 1117	1 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ∉ wnel 3037  ∀wral 3052  ∃wrex 3062   ∩ cin 3889   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  dom cdm 5624  ‘cfv 6492  fBascfbas 21332

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-fbas 21341

This theorem is referenced by:  fbssfi  23812  fbncp  23814  fbun  23815  fbfinnfr  23816  trfbas2  23818  filin  23829  fgcl  23853  fbasrn  23859