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Theorem fbasssin 23659
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) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
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.
StepHypRef Expression
1 elfvdm 6928 . . . . . . 7 (𝐹 ∈ (fBas‘𝑋) → 𝑋 ∈ dom fBas)
2 isfbas2 23658 . . . . . . 7 (𝑋 ∈ dom fBas → (𝐹 ∈ (fBas‘𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧)))))
31, 2syl 17 . . . . . 6 (𝐹 ∈ (fBas‘𝑋) → (𝐹 ∈ (fBas‘𝑋) ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧)))))
43ibi 267 . . . . 5 (𝐹 ∈ (fBas‘𝑋) → (𝐹 ⊆ 𝒫 𝑋 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧))))
54simprd 495 . . . 4 (𝐹 ∈ (fBas‘𝑋) → (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧)))
65simp3d 1143 . . 3 (𝐹 ∈ (fBas‘𝑋) → ∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧))
7 ineq1 4205 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑧) = (𝐴𝑧))
87sseq2d 4014 . . . . 5 (𝑦 = 𝐴 → (𝑥 ⊆ (𝑦𝑧) ↔ 𝑥 ⊆ (𝐴𝑧)))
98rexbidv 3177 . . . 4 (𝑦 = 𝐴 → (∃𝑥𝐹 𝑥 ⊆ (𝑦𝑧) ↔ ∃𝑥𝐹 𝑥 ⊆ (𝐴𝑧)))
10 ineq2 4206 . . . . . 6 (𝑧 = 𝐵 → (𝐴𝑧) = (𝐴𝐵))
1110sseq2d 4014 . . . . 5 (𝑧 = 𝐵 → (𝑥 ⊆ (𝐴𝑧) ↔ 𝑥 ⊆ (𝐴𝐵)))
1211rexbidv 3177 . . . 4 (𝑧 = 𝐵 → (∃𝑥𝐹 𝑥 ⊆ (𝐴𝑧) ↔ ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵)))
139, 12rspc2v 3622 . . 3 ((𝐴𝐹𝐵𝐹) → (∀𝑦𝐹𝑧𝐹𝑥𝐹 𝑥 ⊆ (𝑦𝑧) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵)))
146, 13syl5com 31 . 2 (𝐹 ∈ (fBas‘𝑋) → ((𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵)))
15143impib 1115 1 ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑥𝐹 𝑥 ⊆ (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wne 2939  wnel 3045  wral 3060  wrex 3069  cin 3947  wss 3948  c0 4322  𝒫 cpw 4602  dom cdm 5676  cfv 6543  fBascfbas 21220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-fbas 21229
This theorem is referenced by:  fbssfi  23660  fbncp  23662  fbun  23663  fbfinnfr  23664  trfbas2  23666  filin  23677  fgcl  23701  fbasrn  23707
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